I have an account that is only for private repos and I don't want that account to appear on search engines for my real name.

I also have a public account that I use for contributing to public projects so it could also help to avoid duplicate accounts in search results.

All accounts should be searchable for username/real name with GitHub's search anyway.

Just curious, why not have just a single account?

Your private repo's and contributions are not made public, so there's no "risk" involved.

I think a lot of companies forced this because for a while the api permissions weren't granular enough when dealing with private repos on organizations. They added the "Third-party application access policy" sometime in the last year or two. I might be wrong though.