Had similar issues. The problem is as @Paul Golin and others mentioned: the new Amazon Kids experience restricts websites.
You cannot add the Microsoft Live login sites. The solution is to unrestrict websites, and use the NEW filtering mechanism as follows:
Login as the child
Open Settings (pulldown, go to settings gear), provide parental unlock.
Under "Web Settings" header, there is "Modify Web Browser". Click that, it will ask to use the new Amazon Kids web browser, and sync your settings.
Once it's using the new browser, change the radio option to the bottom choice: "Filtered Websites & Videos". It cannot be on "hand-selected websites & videos".
Now you can sign back into Minecraft, and login with a Microsoft account.
Keep in mind, you just unrestricted the web browser: it now can visit anything on the web. There is an option to add restricted sites, but it's a allow-by-default.
Once you've signed in to Minecraft, I believe you can switch it back to "Hand-Selected websites & videos". But if you ever get logged out, you'll have to repeat steps 1-5 again.
Andy J
Had similar issues. The problem is as @Paul Golin and others mentioned: the new Amazon Kids experience restricts websites.
You cannot add the Microsoft Live login sites. The solution is to unrestrict websites, and use the NEW filtering mechanism as follows:
Login as the child
Open Settings (pulldown, go to settings gear), provide parental unlock.
Under "Web Settings" header, there is "Modify Web Browser". Click that, it will ask to use the new Amazon Kids web browser, and sync your settings.
Once it's using the new browser, change the radio option to the bottom choice: "Filtered Websites & Videos". It cannot be on "hand-selected websites & videos".
Now you can sign back into Minecraft, and login with a Microsoft account.
Keep in mind, you just unrestricted the web browser: it now can visit anything on the web. There is an option to add restricted sites, but it's a allow-by-default.
Once you've signed in to Minecraft, I believe you can switch it back to "Hand-Selected websites & videos". But if you ever get logged out, you'll have to repeat steps 1-5 again.