This only seems to be a problem if you sign out and try to sign in again. I have another tablet that is still logged in and doesn't have any problem staying signed-in.
Also experiencing this on 1.16.101 on Amazon Fire Tablet.
Error message shows before being prompted for credentials.
Tried uninstalling and reinstalling, cleared data, rebooted multiple times, tried using the "remove login data" button in profile screen ... still doesn't work.
Without more detail, I assume this is similar to MCPE-106469
I'm also experiencing this issue. I have included more details on that ticket.