This happened to me this morning and it appears to be a problem with Steam and ESO not validating the account properly, as my non-Steam install works fine.

This is how I worked around it:

• You'll need to know your ESO account name (without the @) and your ESO account password (not your steam PW) and you'll need access to the email you used to set up your ESO website account.

Open Stream, in the Steam menu (top left) choose settings
Choose Account, then in the lower part of the box check "Don't save account credentials"

• NOTE: this will disable "Play Offline" for your other Steam apps, so you'll want to remember how to put this back

Click okay and exit. You've now disabled Steam Auto-Login for ESO.

Now you want to bypass the ESO launcher. You'll need to find your "eso64.exe" file. If you didn't customize, Steam puts it in the "C:\Program Files (x86)\Steam\steamapps\common\Zenimax Online\The Elder Scrolls Online\game\client" folder. Make a shortcut to this on your desktop in case you want to use it in the future.

Run eso64.exe

When the login screen appears, put in your ESO username (without the @) and your password for the ESO website.
If you haven't done this before, it'll block the login waiting for you to put in a code it just emailed you. Do this, and you should get to character select.