Other mistake: The guy who plays Clive in this episode has one of the most appalling accents I have heard. He's apparently from Manchester, but sounds like a sort of cockney South African. His accent slips around all the time; it's very poor. I wouldn't normally submit it, but his is so shockingly bad I doubt anyone would be fooled into thinking he was British.
Suggested correction: I think this may have been deliberate. The episode is almost meant to be ridiculous, with the characters quickly churning out lies on the spur of the moment, and everyone supporting the fake family story without objection.
Plot hole: In this show Daphne's former fiance, Clive, shows up. They had been engaged for several years in England. However, in Series 1, episode 11 (Death Becomes Him), Daphne talks about never having had a serious boyfriend, and goes into some detail about how people tried to set her up with eligible men. As Daphne is open about everything else, there seems to be no logical reason for her to not mention she'd been engaged.
Continuity mistake: Frasier is in the kitchen pouring mini chocolate chip cookies from a bag into a bowl while talking to Daphne. The motion of him pouring (and the sound) doesn't stop while he talks but when the shot changes the bag is on the counter next to him.
Continuity mistake: While Niles, Daphne, and Clive are sitting on the couch talking, the large book on the coffee table goes from closed to opened during the scene, even though no one touches it.
Continuity mistake: When Clive arrives at the apartment, he puts his coat over the front of the arm of the couch. As the scene progresses and Marty tells him about being an astronaut, the coat is suddenly on the back of the arm of the couch, even though no one moved it.
