Sorry, but no - they have not authorized the release of that information, so much as I would not share your email with a random person who asserts they want to use the domain on your site, I can’t do that to them.
I can confirm that login in your screenshot is not one of the members of the paid team to which I refer, but that error message is the same for entering a bad password as for a deleted account, so that isn’t really a good gauge of “did anything get deleted” I can confirm that in this case it is a deleted account, so they didn’t do anything wrong.
I guess next steps are to for my team to contact the current domain owner to ask if they are working with you and authorize the transfer to your control from theirs (their team is fairly large, so it might take a few tries to find someone who is paying attention, but I’m willing to keep trying until we succeed, however of course I cannot promise any timeline for them answering). To hopefully get a good response quickly, what shall I describe you as? I can guess your name from your forum username, but perhaps you work with Agency X or via Person Y’s authority, and I can namedrop to help them understand the additional context, while I say “Matthew is trying to take over control of the website + domain for this name”?
There is another option! You (or someone you work with) can prove control of the domain by adding (at google domains) a TXT record, for the hostname “netlify.isla-beauty.com” with a value of “verified for netlify support Error: Another site is already using this domain”, and that would show that you are authorized. I still have to tell $othercustomer we are removing from their control, since they host the site, but that is something I can tell, instead of asking, so the timeline is just based on how fast you can get the record added and how fast we see your reply.