In "The Greene Murder Case" (about 29 minutes in) someone mentions reading about "The Canary Murder Case". But, in "The Canary Murder Case" (about 21 minutes in) someone mentions that he hasn't seen Vance since "The Greene Murder Case". The studio may not have been sure which order the movies would be released when the dialog was written. Original writer S. S. Van Dine (true name: Willard Huntington Wright) published the first three Philo Vance novels together in a trilogy before separating them into individual books, which might explain the cross-references between the stories.
The phonograph record featured in the climax is supposed to have been custom-made, but the label on screen is an ordinary commercial release on the Victrola (a subsidiary of Victor) label.