In that case, why not use gitlab rather than your service, which isn't compatible with anything?
Also, on github, my name is Fr_Dae, not my real name.
if you have two emails, you can have two github accounts, or two gitlab accounts,
I chose to host my own service as I like to be independent from big tech companies as much as possible. Just accept it. It doesn't need to be compatible with anything but standard git.
Github does not allow multiple accounts even with separate email addresses. I tried it and the second account didn't get activated. I even contacted the support. I was told, it is not allowed.