I see it went wrong even before it got back to ADE ready for your new ID.
Worth separating the two
steps: (1) getting the Adobe ID and (2) applying it to ADE. Removes at least one level of confusion.
It may be there is a Firefox problem with getting a new ID on the Adobe website.
Unlikely to be just Firefox (even Adobe do that much testing), but could be one you your Firefox extensions.
Try Firefox without extensions, or a different browser to sign up for your Adobe ID.
(Or do you already have one to be on this forum? I forget now if the forum has an ID of its own.)
If you give ADE an existing Adobe ID it shouldn't need to go near to any browser to register the ID to ADE.
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You might still need the deregister step ctrl-shift-D as above if ADE is confused.