ooh not to be a history nerd on main but I think it mainly depends on the social status and location of the participants and the exact circumstances of the adoption! (Just to clarify - this is in America? and when you say that it's set in the 1890s, does that mean the adoption happened then and was discovered later, or the adoption is discovered in the 1890s? that might make a difference)
I can recommend this article as a look into adoption laws of the time - you mentioned it was a family adoption, so the class, location and status of your characters is super important here! for example if they're urban middle class fighting out a Big Family Feud they'd be more likely to take it to family court than members of an isolated rural first-generation immigrant community, if that makes sense.
That would also affect their presence in birth registrations - birth records have been kept in some form (usually in church registers) for a really long time, esp. for census and tax purposes, and right around the mid to late 1800s there were calls in America to keep more formalised birth statistics, for medical research and to understand things about the rapidly growing immigrant population. The first standardised birth certificate to be used on a federal level was developed in 1900, though there was legislation in various states about local record-keeping as early as the 17th century. so if the descendants are looking through external records, you'd want to have a specific location and then look up what registration was like there at the time, which you could probably easily find in the context of ancestry research online.
of course, if your descendants are looking through their own family records to find this info, then who they are would also determine what kind of records they keep. What's the literacy like? How are they involved in the legal and economic structures of their community - a lawyer who went to university and wants to form new business partnerships in the city probably has more extensive paperwork than a butcher in a small town. more informally than paperwork, they also might be able to find the info in diaries or letters, or if people who witnessed the adoption process talk to them about it.
asdfkjds this was a lot of words but I hope it was somewhat helpful!