[stumped]

Fire devices, tablets, TV, use same registration process as everything else. - you enter an email on a virtual keyboard, then a password. Most login errors are due to making errors in that. Like upper vs lower case errors.
I cannot recall if password characters get hidden as you type them. That makes it harder to spot if you miss one or repeat one

But from memory it works just like registering a Kindle.
You could though, look at my devices on the web site, to see what is registered , and what is not.
And you can de register a device using the web page.
And you could do through a de register, re register on the tablet. With it online finger device and web site in sync.
Or call in and ask them to watch while you try to register, maybe they can see error messages that you can't.
Could be that login error is really network error.
If the fire will not register and is in warranty they should send you a new one.