I'm coming here to see if you guys can help. When I am playing Xbox 360 games on the XB1 or 360 my account logs out and tells me I have an authentication error. Playing XB1 games is no issue. Has anyone else had to deal with anything like this?
I had this happen several months ago with my account as well. It did occur when I applied two factor authentication, but the Microsoft Representative said it could happen for other reasons (they didn't elaborate). Essentially, the Xbox 360 isn't setup for two factor, so the password sign-in doesn't work, as it is looking for extra information. Passwords on the 360 can only be so many characters, but when logging in with two factor, it adds extra characters that confuses the 360.
Don't worry, if you contact support and explain to them the issue, they can reset it on their end to allow you access. It's fairly easy, but you just need to tell them the code and issue. You'll want to be near both systems (I have both the 360 and One on the same TV, so swapping inputs was what I did). I think you had to delete both accounts and redownload while the representative resets on their end, then they give you a password to input on both and you should be able to play on both again with your 360 library.
I believe I called them about this, but you could probably just chat online with support for this problem.
I had to reset my information recently, but haven't had to use my 360 again. I can try it tonight and let you know if changing any of my password setting has triggered this again.