Several hundreds, at the very least. And new species being discovered all the time. And some will change forms depending on conditions or magical stones.

[He gestures to the Flareon, who hops up into his arms to share a fond headbunt.]

Vonir here began life as an Eevee. Eevees can assume eight different forms. Vonir chose to use a Fire Stone and is now a Flareon. There are also forms connected to water, electricity, ice, plants, psychic power, nighttime, and fairies.

[He is clearly the type of nerd who likes explaining things.]