“Lilmod, if the family isn’t frum, there’s no way for the adopted child to become Jewish even if he wanted to – since he clearly isn’t being mekabel the taryag mitzvos.”
I’m just talking about the metzius – not whether or not it should be that way. I know of families that either weren’t Frum or weren’t so Frum and adopted goyim whom they brought up as Jews who chose not to be Jewish at their Bar Mitzvah.
Regarding whether or not they could have become Jewish if they had wanted to: 1. They could have chosen to be Frum even if their families weren’t. They did go to Frum schools and had some Frum siblings, so they knew what it was about.
2. I don’t know exactly how religious the families were or weren’t. It is possible that they kept some things. They may have kept Shabbos and Kashrus. And halachically, I’m not sure if the family has to keep every single halacha in such a case.