The mouthpiece used is a saxophone mouthpiece (clarinet mouthpieces have cork on the mouthpiece itself), though it still more closely resembles a clarinet because of the likely cylindrical bore (as opposed to the saxophone's conical bore).

well,
a wind instrument then, a hollow tube with holes in it.

Yeah, same basic idea as any woodwind. If he put, say, a trumpet mouthpiece in the carrot instead of a clarinet though, he could of had a crude and pretty shitty carrot trumpet.

I guess all instruments are the same as long as they aren't percussion or string.