The song is a pastiche of early jazz styles, and is perhaps most influenced by the pre-swing early jazz sound called "Dixieland" or "Hot Jazz." Originating in the African-American dance bands of New Orleans, the style is anchored by a fast steady rhythm line (which in your sample is taken by the repetitive chords on the piano, but which would have originally been a banjo) with melodic improvised solos on top by a variety of instruments (the clarinet in your sample).
The minor key pushes it in the direction of what is often known as "gypsy jazz" or "gypsy swing" after the great Roma guitarist Django Reinhardt. (It's worth noting that many people now consider the term "gypsy" to be a racial slur! For that reason, you might prefer to use the alternate genre name, "hot club swing," after the "Hot Club" of Paris). Hot-club uses a guitar as the rhythm instrument, however, so if you're primarily interested in the piano/clarinet sound, that might not be the best reference. (There are, however, hot-club style recordings that do feature clarinet.)
In terms of other similar songs, this has (probably deliberate) echoes of the iconic 1930s Irving Berlin hit Puttin' on the Ritz, and the "Christopher Columbus" section of the Benny Goodman version of Sing, Sing, Sing.