The fact that I logged into the server in Hamburg, does that mean I can use it to determine my identity?

No, for two reasons:

The next server that fits your actual location doesn't have to be the one next to you. You can also use a server hundreds of miles away because the key word is "load balancing" for best performance. You will be assigned to a remote server if it is appropriate.

No one can tell one user from another, so no one will know if you are assigned to the same server or in a different country.