It seems recent updates by Apple have disabled Home Share functions in older (Gen 2 and 3) Apple TVs. I have two older ATVs - one still has full functionality, but the other cannot get on to Home Share.
There is no issue getting into the store or iCloud, but no matter what I try, it throws a password error when trying to activate Home Share on that one ATV.
This appears to be a widespread problem, with dozens of Apple customers reporting the problem in the Apple forum. People are suggesting it is related to two-factor ID or a fault in Apple's authentication servers. I gather this is not the first time this has happened, and it is making a few people contemplate a switch away from Apple for streaming.