You registered on a site to receive free bitcoins as a gift right? So why didn't you get your gift? What kind of a site giving free bitcoins asks for email and not a bitcoin address? You don't even have an address.
It's hard for me to think because in this statement alone I think it's already an answer.
I used an email with random personal data, I remember the password but I can't verify the identity
On the other hand it would be great if the OP linked something else in his email to be able to verify but for it's been over a decade I'm not even sure this email can be recovered especially as far as I know if the email hasn't been used for a long time then it's like it can't be accessed anymore.
Let alone for more than a decade, maybe even for 1 year, the unused email will disappear.
I think giving up is a good thing, not to belittle the seriousness of the OP but this is definitely difficult even if you contact microsoft to prove it.