After reading multiple forums I tried something which worked for me :) At least they charged me and got the confirmation email for the developer program so I made progress :D
What I did were the following:
create a new AppleID;
enrol from Apple Developer App using the new ID (this option was disabled for me until now);
followed on screen instructions (take a photo of my id, insert some personal information);
finished the enrolment but did not introduce any payment information;
SorinC
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Somehow It got the payment information from the payment method set in my AppleID which was logged into the Mac from which I enrolled using the app. So I had one ID which was logged into the mac and iPhone that I have on which I have iCloud enabled and one payment method for it and the Apple Developer App got the payment information from there, not from the newly created AppleID.
Since the money left my account, I assume that in a few days I will have the account and can do the deployments.