You can't push to github without an account. That's how github can validate that you have the rights to push. If that is a problem, host your own gitea instance.
Once you have an account, the way to do it is using ssh keys. Use ssh-keygen (or putty keygen) to generate a key pair, add the public key to your account, and the set up ~/.ssh/config.
If I understand OP's explanation correctly, they're simply trying to make a, possibly selfhosted, copy of a GitHub repo.
In that case the misunderstanding would be in the role of the git command; it being simply a frontend to any git repo, not a client to GitHub.