Our students are unable to sign into or update their Google passwords on the iPad in settings (not the gmail app). I've done some testing and it seems to be something to do with "Only Allow Some Apps" under "Restrict App Usage (supervised only)".
The mail app is on the list to allow but something else seems to be blocking it. The odd thing is, the users receive an email to say they've signed into a new device but it doesn't add the account the iPad.
Just to add a little more info to this:
We have it filed internally under PI-003111 for our own tracking, and through testing discovered the same thing was happening with profiles made in Profile Manager, so this appears to either be working as Apple intended or an issue on Apple's side; we're not sure which it is just yet.
There is an AppleCare Ticket open under 100043508405.
As @RLR mentioned, to get it to work with Google accounts, you'd need to allow the app com.apple.social.SLGoogleAuth.
For Yahoo: com.apple.social.SLYahooAuth
For Outlook: com.apple.dataaccess.dataaccessexpress.framework.EASOAuthService