Can an assigned server compromise my anonymity ?

Can the fact, that I'm logged in to, say, a server in Hamburg, be a hint to where I live and be used to identify me?

No, for two reasons:

• The next best suited server near your real location is not necessarily the one, that is really located next to you. It could also be a server hundreds of miles away, because the keyword is 'load balancing' for maximum performance. If a server farer away is better suited, then that's the one you will be assigned to.
• Nobody can tell one user from the other - so nobody will know if you have been assigned a server or logged in to the same server from a different country.