Short answer
Not in the same way. The Qur'an was learned by heart and written down while the Prophet (ﷺ) lived. It was gathered into one volume within two years of his death, and within twenty years copies of that one text went to every province. Other faiths kept their books with real care, and Islam honours that. But their own sources tell of losses, rival versions or late fixing. Only one school of the Rig Veda really survives. The Jains lost older works of theirs, the Purvas. A Buddhist chronicle tells of a council that changed the order of the texts. The Kojiki's own preface says the court sifted records full of errors. No one knows for sure who wrote the Tao Te Ching. Scholars still disagree which of two early copies of the Sikh scripture has the original form of one hymn. And the Analects are not named until centuries after Confucius. The Qur'an promised that God (﷾) would guard it, and its record bears that out.
Full explanation
The measure
The Qur'an was kept in two ways at once. Each passage was learned by heart and written down by scribes as it came, and Gabriel went over the whole text with the Prophet (ﷺ) every year. Within two years of his death it was gathered into one volume, from the written pieces and the memories of men. Within twenty years, copies of that volume went to every province. So the questions for any scripture are the same. Was it written down while its teacher lived? Was it gathered while those who heard it were alive? And has one text come down?
Abu Hurayra reports that Gabriel went over the Qur'an with the Prophet (ﷺ) once every year, and twice in the year he died.
Zayd ibn Thabit reports that after many who knew the Qur'an by heart were killed at Yamama, Abu Bakr asked him to gather the Qur'an into one volume, reminding him that he used to write down the revelation for the Messenger of Allah. Zayd collected it from written pieces and from the memories of men, and the finished copy passed from Abu Bakr to Umar and then to Hafsa.
Anas reports that Hudhayfa ibn al-Yaman, alarmed at how the soldiers of Syria and Iraq differed in reciting the Qur'an, urged Uthman to save the community before it differed over its Book as the Jews and Christians had. Uthman borrowed the collected sheets kept by Hafsa, had Zayd ibn Thabit and three men of Quraysh copy them, telling the three that wherever they differed with Zayd they should write in the dialect of Quraysh, in which the Qur'an was revealed, returned the sheets to her, sent a copy to every province, and ordered all other written Qur'anic material to be burned.
What their own sources say
Each tradition's own records tell its story. The Rig Veda was handed down with great care, but it survives in only one school, and the others are lost. Both main Jain groups say the Purvas, older works than those that survive, were lost until no one knew them. A Buddhist chronicle of Sri Lanka says a rival council of monks changed the order of the texts. The Kojiki's preface says the court found the old records full of errors and sifted them. Sima Qian, an early Chinese historian, gives more than one account of who wrote the Tao Te Ching. Scholars still disagree which of two early copies of the Sikh scripture has the original form of one hymn. And scholars note that no one quotes or names the Analects before the second century BCE, about three hundred years after Confucius.
Jamison and Witzel write that the Rig Veda has come down basically in only one extremely well preserved school, that of Shakalya, whose word-by-word analysis of the text (the Padapatha) is also handed down; the Bashkala and Mandukeya collections are lost. Their table of Vedic texts lists the surviving collections: one of the Rig Veda, three of the Sama Veda, six of the Yajur Veda (one of them partial) and two of the Atharva Veda, which they say differ considerably from each other. The table marks other texts as lost, such as the Paippalada school's Brahmana, which has no surviving text.
“we are told by the Svetambaras, as well as the Digambaras, that besides the Angas, there existed other and probably older works, called Purvas, of which there were originally fourteen. The knowledge of these Purvas was gradually lost, till at last it became totally extinct.”
“The Bhikkhus of the Great Council settled a doctrine contrary (to the true Faith). Altering the original redaction they made another redaction. They transposed Suttas which belonged to one place (of the collection), to another place; they destroyed the (true) meaning and the Faith, in the Vinaya and in the five Collections (of Suttas).”
“I hear that the chronicles of the emperors and likewise the original words in the possession of the various families deviate from exact truth, and are mostly amplified by empty falsehoods.”
Sima Qian's chapter on Laozi says he was Li Er, also called Dan, a man of Chu who kept the archives at the Zhou court. Laozi cultivated the Way and virtue, and his learning aimed at hiding himself and having no name. When he left Zhou, the keeper of a pass asked him to write a book. So he wrote a book in two parts, of some five thousand words, on the meaning of the Way and virtue. Then he left, and no one knows where he ended. The chapter then gives other accounts. Some say it was Lao Laizi, also a man of Chu, who wrote fifteen chapters on the uses of the Taoist teaching. It records that the Grand Historian of Zhou, also named Dan but written with a different character, met Duke Xian of Qin long after Confucius died. Some say this Dan was Laozi, some say not, and no one knows whether it is so. It calls Laozi a gentleman who lived hidden from the world.
Says Guru Arjan and his scribe Bhai Gurdas completed the first edition, the Adi Granth, in 1604 and installed it in the Golden Temple at Amritsar. Its hymns are chiefly by six of the Gurus, with others by Hindu bhakti saints and the Sufi Sheikh Farid. Guru Gobind Singh later added the hymns of Guru Tegh Bahadur to make the final edition. The article reports a scholarly debate over two early manuscripts: at one place the Kartarpur manuscript has only the two opening lines of a hymn of Guru Arjan, followed by blank folios, while the Banno manuscript, dated 1644, has the whole hymn. Scholars disagree which reflects the original.
Reports a second-century CE account by Ban Gu that the Analects was compiled by the first and second generations of Confucius's disciples and then handed down privately for centuries. Notes that the classicist Zhu Weizheng argued in 1986 that the Analects is neither quoted nor named before the second century BCE, the century in which its text was fixed, and that excavated copies date from the first century BCE. Suggests the Analects may in part be a selection from a larger and more varied body of Confucius-related material.
A Book that asks to be tested
The Qur'an does not ask to be taken on trust. It says God (﷾) sent it down and will guard it. It says that had it come from anyone but God (﷾), people would find much discrepancy in it. And it challenges anyone in doubt to produce one surah like it. The page "How can we tell if a holy book really comes from God?" sets out how to test any scripture.
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
“Do they not consider the Qur'an (with care)? Had it been from other Than Allah, they would surely have found therein Much discrepancy.”
“And if ye are in doubt as to what We have revealed from time to time to Our servant, then produce a Sura like thereunto; and call your witnesses or helpers (If there are any) besides Allah, if your (doubts) are true.”
Perspectives
Hindu: the Veda has been kept word for word
Hindu tradition holds that the Vedas were handed down by memory with great care. Scholars say the Rig Veda that survives is very well kept, like a tape recording passed on without a word changed.
Scholars say the Rig Veda that survives is extremely well kept. They compare it to a tape recording, usually handed down without a single word changed.
Rebuttal
That is true: what survives was kept with great care, and Islam honours that. But care for what survived cannot bring back what was lost. Of the Rig Veda's old schools, only one text really remains, and Patanjali speaks of many branches of the Veda. The Qur'an was kept whole, as one text, by memory and writing together, as God (﷾) promised to guard it.
Buddhist: the monks kept the texts by memory
Theravada Buddhists hold that monks handed down the Buddha's teaching by memory. Later they wrote it down in Sri Lanka, so that it would not be lost.
The monks handed down the three Pitakas, the Buddhist scriptures, by memory. They wrote them in books only so that the teaching would last.
Rebuttal
The same chronicle says that a rival council of monks changed the original order and moved teachings from one place to another. So memory did not keep one text. It kept several, and each school says its own is the true one. The Qur'an has one text, memorised and written from the start, and God (﷾) promised to guard it.
Jain: the surviving Angas carry the teaching
Jains hold that the teaching of Mahavira was handed down by memory, and written down later at a council in Valabhi. Both main Jain groups agree that the oldest works, the Purvas, were slowly lost.
The Angas, the Jain scriptures that survive, were written down at Valabhi. They still carry the Jina's teaching, even though the Purvas are lost.
Rebuttal
They may carry much. But the tradition itself says the Purvas were lost until all knowledge of them was gone, and nothing can bring them back. The Qur'an lost nothing. It was memorised and written in the Prophet's (ﷺ) lifetime, and gathered within two years of his death.
Shinto: the Kojiki kept the old words
The Kojiki, the oldest Japanese book, was presented at court in 712. Its preface says a court servant named Are was ordered to learn by heart the family lines of the emperors and the words of earlier times. These were then written down.
Are learned the old words by heart at the emperor's command, and the Kojiki preserved them in writing.
Rebuttal
The same preface says the court found the old chronicles and the families' records full of errors. It had them sifted, "falsehoods being erased and the truth determined". By its own account the Kojiki is a court's careful choice from mixed sources. It never claims to be God's (﷾) word. The Qur'an claims to be His word and says He will guard it.
Daoist: Laozi wrote the book at the pass
Daoist tradition holds that Laozi wrote the Tao Te Ching as he left the Zhou court. The keeper of a border pass asked him to write it. It has been chanted as scripture ever since.
When Laozi left Zhou, the keeper of the pass asked him to write a book. So he wrote the five thousand words of the Tao Te Ching.
Rebuttal
That is the first account Sima Qian gives, but he goes on to give others. Some say the author was Lao Laizi, some say a later man named Dan, and no one knows which is true. Even the book's author is uncertain. The Qur'an was recited by the Prophet (ﷺ) in public, written down by his scribes and memorised by his Companions while he lived.
Sikh: the Guru compiled the scripture himself
Sikhs hold that their scripture was written down by the Gurus themselves. Guru Arjan and his scribe Bhai Gurdas finished the Adi Granth in 1604, and Guru Gobind Singh gave it its final form.
Guru Arjan compiled the Adi Granth himself in 1604, so the Sikh scripture was fixed by its own Guru.
Rebuttal
Islam agrees that a scripture written down in its teacher's own lifetime is a strength. The Qur'an was written by the Prophet's (ﷺ) own scribes, and memorised at the same time. Even so, early Sikh copies differ. At one place, the Kartarpur manuscript has two lines and blank pages, where the Banno manuscript of 1644 has a whole hymn. Scholars still disagree about which is the original. The Qur'an has one text. It says every word is God's (﷾), and invites anyone to test it.
Confucian: the disciples recorded the Master's words
Confucians hold that the Analects record the Master's sayings as his followers remembered them. They were handed down with care, generation after generation.
The Analects were compiled by Confucius's own disciples and their students, and handed down with care.
Rebuttal
That account comes from Ban Gu, who wrote more than five hundred years after Confucius. Modern scholars note that no one quotes or names the Analects before the second century BCE, when its text was fixed. It may be a selection from a larger set of sayings. And Confucius never claimed his words came from Heaven. The Qur'an was written down in the Prophet's (ﷺ) lifetime and gathered within two years of his death.
Chinese tradition: customs, not scripture
Chinese popular religion has no single scripture. Its ways, such as honouring the ancestors and the festival of the seventh month, are kept as family custom. Parents hand them down to their children.
Our tradition needs no scripture: the customs are handed down in families, which is how they are kept.
Rebuttal
Customs passed on by word of mouth change from place to place and from family to family. And none of them claims to be God's (﷾) word. The Qur'an does. It was written down by the Prophet's (ﷺ) scribes and memorised at the same time, and God (﷾) promised to guard it.
The Islamic perspective
God (﷾) revealed the Qur'an and promised to guard it. It was kept in memory and in writing from the start, and gathered as one text while those who heard it from the Prophet (ﷺ) were alive. The other traditions' own sources tell of losses or rival versions.
The Qur'an promises that God (﷾) Himself will guard it.
It was memorised and written in the Prophet's (ﷺ) lifetime and gathered within two years of his death. Within twenty years it was copied for every province.
The other traditions' own sources record lost works, lost schools or rival versions.
The Qur'an asks to be tested. It says it has no discrepancy, and challenges anyone to produce a surah like it.