In the US today, is there any defined modern family?
Clearly, marriage for Caucasians in the US up through WW II (1940s) was largely defined by the Christian religion as interpreted in England, Europe, and the colonies. (The Jewish tradition followed along similar lines.) Sex before marriage was sometimes accepted, marriage was heterosexual and intended lifelong, divorce or annulment very rare. Families were extended clans of grandparents to great-grandchildren. and uncles, aunts, nephews, nieces, and cousins made up an extended family with varying degrees of closeness. Extramarital sex was frowned on and kept hidden, as was homosexual desire or activity.
Black, Native American, and other sub-cultures have their own history of the nature of family, not discussed here.
The first change came in the 1950s, and it was due to modern transportation and marketing, not religion. The "nuclear family" came to refer to parents and their children set off separately from the clan-like extended family. As men came back from the wars (WW II and Vietnam), migration to California increased, and economic prosperity increased, the myth of the nuclear family - a husband earning a living, a wife raising children with little outside help, and an average of 2.3 children per family was born.
The next change was due to the invention of reliable birth control in the 1960s. This opened up new opportunities for sexual activity, the divorce rate skyrocketed, and divorce for emotional reasons became acceptable. This meant that marriage was no longer likely to be lifelong, and children might not be in a household with both parents through their childhood.
Starting in the 1970s, and increasing through today, a single income became not enough to support a family. That, combined with divorce rates, create single-family homes with a working mom (or dad), families with two working parents, and blended families. This is an economic influence, not a religious one.
More and more, children did not have the role model of lifelong marriage in their parents. Now, parenthood without marriage is increasing rapidly.
Public openness on homosexuality for men and women changes our definition of family as well. This is a cultural issue, not a religious one.
Children exposed to abuse, especially sexual abuse, tend to grow up to have non-traditional patterns of sexual activity. This changes family patterns, too.
Conclusion: Since the 1950s, religion has been a small influence on family definition.