Apparently, "Kobe" is used to describe many things, and, given that a prominent place like the burger bar is still selling their burgers 10 years later, their doesn't appear to be a lot of trademark police chasing people down.
"Kobe beef" is trademarked in Japan and is legally protected there, but not in the US, where anyone is free to use it as they like. You can still prove a beef cut has come from the real source via certification.
These things usually get codified in trade agreements. It seems that Japan hasn't been particularly concerned about protecting Kobe beef internationally, likely because there are so few exports simply due to production limits.
http://travel.yahoo.com/p-interests-23233092