@Franciszek It sounds as though the problem is with Github or your credentials, not with Netlify. With a Netlify subdomain you will not have any DNS entries, so there should be no interaction.
Can you gain access to your Github account separately, that is, without going through Netlify? Once you establish that connection, check your profile settings to ensure that you are using the e-mail account you think you are using, and check the Applications page to see that Netlify is connected.
FWIW, I just logged out of my Github account and logged back in, and it didn’t ask for my e-mail address, it gave me the option of entering my user name or my e-mail address.