Hi Thomas,
As @electronicsboy said, replacing the battery does not affect the Touch ID function; probably the reason the two seem to be associated is that there is always the possibility of damaging a connector or cable when removing and replacing the screen. Done correctly, it won't hurt your home button.
However, it is true that Apple is now serializing the batteries so that replacing the battery will give you a non-genuine battery warning ***even if it is a genuine Apple battery***. The Apple self-repair program is supposedly their remedy for that situation, but of course you have to buy the battery from them and use their equipment in order for that to work.
Users have discovered that transferring the BMS, or Battery Management System board from the old battery to the new replacement battery will allow you to get around the issue of the battery warning. Basically you have to cut the BMS board off your original battery and spot weld it to the new battery, then use a device programmer to reset the battery health and cycle count numbers.
If you go that route, there are plenty of places selling replacement batteries without the BMS for just such a repair, or you can just cut the BMS off the new battery. Here's a video stepping you through the process.
[link|https://www.youtube.com/watch?v=1WA0LoJRjZU|iPhone SE 2(2020) “Important battery message” fix - battery replacement - YouTube|new_window=true]