Apple TV won't login to icloud account

That is a 2nd gen. When using 2 factor authentication on an older model you need to enter the password and code together, and there is a time limit. To make the best use of that time request the code after the password has been entered.

I've got the same model, running 6.2.1 - the most updated (and final) OS for that generation.

Personally, I'm entering the iCloud password, pausing, requesting the code via iPhone, adding the 6 digits to the end of the password, and then submitting - in attempt to avoid the assumed 30/60 second time limit for the two-factor code.

That said, I still get the "try again later" message every single time I attempt it.

The last time I was trying this same method over and over (and it also never worked) was on an old Gen1 that I inherited from my father (I just wanted to use it as a music streamer - airplaying from my iMac and only connecting speakers - for the dining room) and was informed by Apple that there is ZERO support for the original Apple TV, which won't accept the 6-digit add-on password to sign in for home sharing.

And now - back to the ATV2 - when trying to get photos to appear in the screensaver, I tried to sign out of iCloud on the ATV... which was obviously a huge mistake, as it appears that the ATV2 is in the same boat as the ATV1, and no longer allows sign in due to two-factor authentication.

Actually, out of frustration, I just hard reset (pulled power) and let it boot back up... upon start-up, it triggered the two-factor on my iPhone, I tried logging in again and - to my elated surprise - it worked!

I guess Apple hasn't completely given up on Gen 2s yet! :)

Are you entering the password and then requesting the code? If you are trying to enter the password and code all at once then you are probably exceeding the time limit.