Do legacy surnames have to be unique?

Yes, legacy names are unique per server and all your alt toons will have the same legacy name on that server.

If I picked "StackExchange" as a legacy name on one server, no one else can pick that name on that server. But if they moved to another server where it wasn't taken, then it's up for grabs.

Better to identify people's alts with.

Don't believe me? Take a look here and here.

No. This restriction was removed during the latest server merges, with Patch 1.3.7. In the first server merge I was required to change my Legacy name due to the constraint, but after the second merge I was able to change it back.