Short answer
Most of your family life stays, and your duty to your parents grows. You keep your family name, your language, your meals with the family and your respect for elders. Islam commands kindness to parents even if they never become Muslim. What changes is worship. Prayer, offerings and bows of worship go to God (﷾) alone, so a Muslim does not pray to anyone else at a household shrine. Some details depend on the case, such as a bow of greeting, a family funeral at a temple or a New Year shrine visit. For those, ask a qualified scholar about your own situation. Islam is a whole way of life, and caring for your family is part of it.
Full explanation
What stays
Islam does not ask you to leave your family or your people. The Qur'an calls the differences of languages and colours among the signs of God (30:22). It says that He made people into nations and tribes so that they may know one another (49:13). So your family name and your language stay yours. Meals stay too. The word itadakimasu before eating is a word of thanks. The Qur'an also tells believers to eat of the good things God (﷾) has provided and to be grateful to Him (2:172). A Muslim begins by mentioning the name of Allah (﷾), as the Prophet (ﷺ) taught a boy at his table (Bukhari 5376). Food is eaten halal, which means lawful as Islam defines it. Respect for elders also stays, and it deepens: the next block takes it up.
“And among His Signs is the creation of the heavens and the earth, and the variations in your languages and your colours: verily in that are Signs for those who know.”
“O mankind! We created you from a single (pair) of a male and a female, and made you into nations and tribes, that ye may know each other (not that ye may despise (each other). Verily the most honoured of you in the sight of Allah is (he who is) the most righteous of you. And Allah has full knowledge and is well acquainted (with all things).”
“O ye who believe! Eat of the good things that We have provided for you, and be grateful to Allah, if it is Him ye worship.”
Umar ibn Abi Salama reports that, as a boy in the Prophet's (ﷺ) care, his hand roamed around the dish while he ate. The Prophet (ﷺ) told him to mention the name of Allah, to eat with his right hand, and to eat from the part of the dish nearest to him. He says he kept to this from then on.
Your duty to your parents grows
The Qur'an puts kindness to parents beside the command to worship God (﷾) alone. It forbids even a word of contempt to a parent in old age. It teaches the child to be humble and to pray for them (17:23-24). Kindness to them does not end if they are not Muslim. If they press a child to join others with God (﷾) in worship, the child does not obey them in that. Yet the child is told to keep them company in this life with justice and consideration (31:15; 29:8). Asma's mother was still a polytheist and had come to visit her. The Prophet (ﷺ) told Asma to keep good relations with her mother (Bukhari 2620). The Qur'an allows kindness and justice towards those who do not fight you for your faith or drive you from your homes (60:8). Family love is a duty in Islam, not a thing to be given up.
“Thy Lord hath decreed that ye worship none but Him, and that ye be kind to parents. Whether one or both of them attain old age in thy life, say not to them a word of contempt, nor repel them, but address them in terms of honour. And, out of kindness, lower to them the wing of humility, and say: "My Lord! bestow on them thy Mercy even as they cherished me in childhood."”
“"But if they strive to make thee join in worship with Me things of which thou hast no knowledge, obey them not; yet bear them company in this life with justice (and consideration), and follow the way of those who turn to me (in love): in the end the return of you all is to Me, and I will tell you the truth (and meaning) of all that ye did."”
“We have enjoined on man kindness to parents: but if they (either of them) strive (to force) thee to join with Me (in worship) anything of which thou hast no knowledge, obey them not. Ye have (all) to return to me, and I will tell you (the truth) of all that ye did.”
Asma, daughter of Abu Bakr, reports that her mother, who was still a polytheist (mushrika), came to visit her in the Prophet's (ﷺ) lifetime, hoping for her daughter's kindness. Asma asked the Prophet (ﷺ) whether she should keep good relations with her mother. He said yes: keep good relations with your mother.
“Allah forbids you not, with regard to those who fight you not for (your) Faith nor drive you out of your homes, from dealing kindly and justly with them: for Allah loveth those who are just.”
What changes: worship goes to God (﷾) alone
The one change is whom worship is given to. The Qur'an says that the places of worship are for Allah (﷾), so do not invoke anyone along with Him (72:18). Prayer, offerings and bows of worship therefore go to God (﷾) alone. In practice, a Muslim does not pray or make offerings to the kami at the kamidana, the household shrine for kami. Kami are the sacred powers Shinto honours, and Shinto priests say they are not "gods" in the Western sense. The same holds at the mitamaya (also called soreisha), the Shinto shrine for the family's ancestors. It holds at the butsudan, the Buddhist home altar, and at Obon, the Buddhist observance for the family dead. These belong to different traditions, and the principle is the same for each. The Qur'an also teaches trust in God (﷾) for what is good and safe in life: whoever trusts in Allah (﷾), Allah (﷾) is enough for him (65:3). So a Muslim looks to God (﷾) for protection and good fortune, not to omens or fortune slips. The page "What does Islam say about superstition, omens and fortune-tellers?" explains why.
“"And the places of worship are for Allah (alone): So invoke not any one along with Allah;”
“Thy Lord hath decreed that ye worship none but Him, and that ye be kind to parents. Whether one or both of them attain old age in thy life, say not to them a word of contempt, nor repel them, but address them in terms of honour. And, out of kindness, lower to them the wing of humility, and say: "My Lord! bestow on them thy Mercy even as they cherished me in childhood."”
“And He provides for him from (sources) he never could imagine. And if any one puts his trust in Allah, sufficient is (Allah) for him. For Allah will surely accomplish his purpose: verily, for all things has Allah appointed a due proportion.”
Three questions that depend on the case
Some matters are not settled by a short answer. The first is a bow used as a greeting. The second is a family funeral at a temple. The third is joining the family on a New Year shrine visit. The principle is the same in all three: worship is for God (﷾) alone, and family ties and kindness are kept. How it applies depends on what is done and what is meant. This page gives general information and no ruling on them. For your own situation, ask a qualified scholar. It helps to ask one who knows both Islam and your family's customs.
“"And the places of worship are for Allah (alone): So invoke not any one along with Allah;”
“"But if they strive to make thee join in worship with Me things of which thou hast no knowledge, obey them not; yet bear them company in this life with justice (and consideration), and follow the way of those who turn to me (in love): in the end the return of you all is to Me, and I will tell you the truth (and meaning) of all that ye did."”
A whole way of life, with your family in it
Islam is a whole way of life: belief, worship, family and character together (2:208). It does not ask you to take its word on trust. It invites you to look at the Qur'an and test it (2:23). Then ask whether Muhammad (ﷺ) is the last of the prophets, as the Qur'an says (33:40). The Qur'an tells you how to treat your family along the way (31:15).
“O ye who believe! Enter into Islam whole-heartedly; and follow not the footsteps of the evil one; for he is to you an avowed enemy.”
“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.”
“Muhammad is not the father of any of your men, but (he is) the Messenger of Allah, and the Seal of the Prophets: and Allah has full knowledge of all things.”
“"But if they strive to make thee join in worship with Me things of which thou hast no knowledge, obey them not; yet bear them company in this life with justice (and consideration), and follow the way of those who turn to me (in love): in the end the return of you all is to Me, and I will tell you the truth (and meaning) of all that ye did."”
Verse by verse
Qur'an 60:8 — Surat al-Mumtahana (She that is to be examined), verse 8
“Allah forbids you not, with regard to those who fight you not for (your) Faith nor drive you out of your homes, from dealing kindly and justly with them: for Allah loveth those who are just.”
Cited for
Muslims are told to be kind to people of other faiths. So a Muslim may join the family's shrine and temple rites as an act of kindness.
The response
60:8 allows dealing kindly and justly. It does not say to share in anyone's worship. 31:15 puts both in one verse. Keep company with justice and consideration, and do not obey in worship.
Where it lands — Kindness and care are commanded. Sharing in worship is not part of that kindness.
Sahih al-Bukhari 2620 — Kitab al-Hiba (Gifts), hadith 2620
Asma, daughter of Abu Bakr, reports that her mother, who was still a polytheist (mushrika), came to visit her in the Prophet's (ﷺ) lifetime, hoping for her daughter's kindness. Asma asked the Prophet (ﷺ) whether she should keep good relations with her mother. He said yes: keep good relations with your mother.
Cited for
Asma's mother was a polytheist, yet the Prophet (ﷺ) told her to keep close ties, so staying close includes the family's rites.
The response
The hadith tells Asma to keep good relations with her mother. It says nothing about taking part in her mother's rites.
Where it lands — Ties are to be kept. The hadith does not make worship part of them.
Qur'an 72:18 — Surat al-Jinn (The Jinn), verse 18
“"And the places of worship are for Allah (alone): So invoke not any one along with Allah;”
Cited for
A bow or a small offering at the family shrine is only habit and thanks, not worship.
The response
Prayer and offerings at a family shrine are worship of the kami or the ancestors it is set up for. A Muslim gives worship to God (﷾) alone (72:18; 6:162-163), so he makes none there. A bow used only as a greeting is a separate question. So are a temple funeral and a New Year visit. This page gives no ruling on those.
Where it lands — Prayer and offerings at the family shrine are worship, so a Muslim makes none there. A greeting bow is a question for a qualified scholar.
Perspectives
Shinto: custom is family, not creed
A family that keeps shrine, temple and household customs may see them as ways of belonging and gratitude, not a creed. To step back from them can then feel like stepping away from the family.
Joining the family at the shrine or temple is respect for the family, not worship. And the Qur'an itself commands respect for parents.
Rebuttal
Respect for parents is a duty, and it grows with faith. The same verses put worship of God (﷾) alone first. A Muslim keeps company with parents with justice and consideration, and does not obey them in worship.
Islam commands kindness to people of other faiths, so joining their rites is only kindness.
Rebuttal
Kindness and justice are commanded, and 60:8 allows them. The hadith tells Asma to keep good relations with her mother. Neither says that she joined her mother's worship.
Languages and nations are God's own signs, so the customs of a nation should be respected as they are.
Rebuttal
Language, family name and nation stay. Languages are among His signs, and nations exist so that people may know one another. Honour with God (﷾) goes to the most righteous, not to a custom, and worship stays for Him alone.
If you step back from the family's rites, the family will feel rejected.
Rebuttal
Islam does not ask anyone to break ties. Keeping ties and kindness is a duty, even with parents who are not Muslim (31:15; Bukhari 2620). You can decline an act of worship and still care for your family.
The Islamic perspective
Islam keeps family ties, language, meals and respect for elders, and raises the duty to parents. It changes one thing: prayer, offerings and bows of worship go to God (﷾) alone.
The Qur'an commands kindness to parents right after the worship of God (﷾) alone. It forbids even a word of contempt to them in old age.
If parents press a child to join others with God (﷾) in worship, the child does not obey in that. The child still keeps them company with justice and consideration.
The Prophet (ﷺ) told Asma to keep good relations with her mother, who was still a polytheist.
The Qur'an allows kindness and justice toward others. It names those who have not fought you for your faith or driven you from your homes.
Languages and colours are among God's (﷾) signs. Nations were made so that people may know one another, and the most honoured with Him is the most righteous.
A Muslim eats of the good things God (﷾) has provided and gives thanks to Him. He begins a meal by mentioning the name of Allah (﷾).
Places of worship are for Allah (﷾) alone. So prayer and invocation are not given to anyone along with Him.
A Muslim puts trust in God (﷾) for protection and good fortune.