Can someone explain "Log In With Apple?" What exactly is the site I'm accessing seeing? What can they access from that?
icloud - Issue with account.apple.com - Privacy Consent and Account Migration - Ask Different
PSA: Your Apple ID will be renamed ‘Apple Account’ next month
Apple ID is no more on web, long live Apple Account
Videos
Can they access anything personal? My AppleID? How does the login process work with them?
If they use something like Google Analytics, is any of my data shared?
On the assumption that you have no apple device to allow you to use online support via entering a valid serial number, you may have to create a brand new Apple Account online so that you can use Apple Support to get a call back and then have them help you with the main account.
Apple account web pages widely work with non-apple browsers on windows and many unix systems, so unless you have a very unusual browser set up, the problem is likely with the account online and not your computer being unable to log in. It's a bit of a catch-22 to need to log in to get help with an account you can not log-in-to.
Perhaps a friend would open a ticket to assist if you are not convenient to an Apple Store?
Also, is there any chance your IP address or account may be flagged as evading store policies or getting extra scrutiny? If so you might have to start over with what Apple considers a legit registration. If not, making a new legit account and using Apple support should be straightforward.
The short answer to this question is that you're not able to fix this.
This problem is on their end, seems to affect people who don't have Apple devices and created an account for Apple TV or Apple Music (the few services they have that don't require Apple devices) and, as evidenced by this 1 year old reddit thread with 934 comments that continues to be active to this date
This makes me conclude this is by design and not getting fixed any time soon.
Some of the comments suggest that it's an intentional conspiracy to to force people to purchase Apple products. Lots of comments suggest that the process works if you use Firefox and set the user agent to mimic an iPad.
This is currently the most viable way to try to make an Apple account, and would also seem to support the "it's intentional" theory.
Even more incredibly, comments also report being able to make an account by spamming the button thousands of times.
In short, this is a known problem that is not getting fixed and there is nothing any of us can do about it.