araquen solved it:

Okay, so I got Youtube to work. These are the steps I took:

1) I unlinked my YouTube account from my Gmail Account.

2) I changed me email in my YouTube account to my gmail address.

3) I logged in to myYouTube account using my YouTube log in.

4) I linked my YouTube Account to my Gmail Account.

5) I logged into my YouTube app using my Gmail Account information.

I am not sure if this is the true fix, but my thought is that if I force everything to look at the same data, some part of the login process has simply got to stick.

I cannot guarantee that this will work consistently. My understanding is that most fixes that have popped up on the internet have been temporary at best. I also did not try logging in to an unlinked account.

I would say that the root cause of this issue is the kludgy link process between gmail and YouTube and the app’s inability to navigate the kludge. I may be wrong.

via Apple – Support – Discussions – Cant sign in to YouTube App ….