So there's a few threads on Microsoft forums giving ideas for overcoming error 2603 & 2604 sign in errors and they all miss the reason and leave us all frustrated!
When any of the Office 365 apps fail to allow sign-out and sign back in with either just a blank white popup window showing nothing OR with a basic "Failed to Sign In error" loop, it is 99% caused by your Internet Properties settings on your device not having a simple check mark against the "Use TLS 1.2" option.
For Windows systems: navigate to Internet Options in the Control Panel. In the Advanced tab, enable the ‘Use TLS 1.2’ option.
[I tried to upload a graphic for you in both .JPG and PNG, but no surprise, both times, the image uploader failed].
In other device systems, you need to find general Internet Options and look for supporting TLS at least version 1.2
ps. It would be nice for a forum Admin to pin this post to the other locked threads so that customers can at least find a proper answer to this widespread problem.