Sometimes it happens to me, haven't been able to work out a trigger. Visually it gets stuck on the character logging out and never makes it to the selection screen. Since it gets stuck there's no popup screen to enter the details of what went wrong.

I had a couple of hiccups on PC EU yesterday the first time I logged into the game after maintenance.

The first time my computer mostly froze up when I was trying to log out to character selection while in the Vivec City Outlaws Refuge; the picture was frozen-- the first clue that something was up was when I noticed that the ouroboros in the lower right corner of the screen wasn't turning-- and the computer display wouldn't respond to Alt-Tab or Ctrl-Alt-Delete, yet my triangular mouse pointer was still able to move around as though everything were okay. I had to hold in the power button to shut down the computer and restart.

When I got back in after that, I was able to continue logging in and out of my characters one by one to get their mail from the crafting assistants, but after I got the mail on my final character I got an "unexpected error" or some such message when I tried to log out and return to character selection. It booted me from the server, back to the server selection screen, which was okay because that's where I'd wanted to go, anyway. I switched servers and was able to successfully get the mail on all of my PC NA characters without any further incidents.

Edit: I meant to ask, what platform or OS are you on? On my Macbook Air I used to have similar cases of the game closing unexpectedly when logging out of a character so I could switch characters, or after clicking the option on the character selection screen to go back to the server selection screen so I could switch servers. For a while-- last year or before-- that was happening so consistently that it was making it impossible for me to play ESO on my Macbook Air. Fortunately, those problems seem to have been fixed many months ago, or else they went away after some macOS update. Hopefully they haven't started happening again.

been happening to me for months now

Are you by chance using razer app? I had uninstall mine to fix the problem. it cause weird conflict with eso and makes the game close when trying to go to character select screen.