Only two of the Superpets - Krypto, and Beppo the Supermonkey (used for the proof-of-concept rocket before the one that sent Kal-El to Earth) - were Kryptonian. The others were Streaky, an Earth cat given powers by accident, Comet a centaur-turned-horse-sometimes-turned-human who had a thing for Supergirl, and Proty II, a Protean (basically a sentient, shapechanging blob).
Interestingly, the Proteans were later given rights as a sentient race, making Proty's original status as a pet...problematic.