When Frasier tries to get the receptionist to tell him about the club, and to gain entry, he gets denied. The receptionist says that even though he has an invitation, he's not on the list and could have stolen the invitation, which is only sent to people on the list. Then Niles walks in and Frasier calls him Cam Winston(the name on the invitation). Niles pretends to be Cam and says Frasier is his guest. At an exclusive place like this, where it's so hard to get in, the receptionist would have surely asked for an ID from Niles to prove he is Cam Winston.
When Martin and Daphne are watching TV together, they keep referring to the show as "Rockford", but in reality it was called "The Rockford Files". However the show was often referred to casually or colloquially as simply Rockford.
In the first scene, as Frasier is hanging up his coat, Kelsey Grammer is seen wearing black sneakers with his suit with no mention or explanation of it in the show. Frasier Crane would never commit such a faux pas. Obviously Kelsey Grammer wore these for comfort, not intending the shoes to be seen on camera.