What is this style of painting called?

Yes, it’s posterization.

(and I’m an Actual Professional Art Teacher, so you can trust me).

But why do you need to come up with a name for it? Do you want to commission someone to paint your dog in that style? Or google other art in that style?

Have you tried searching for “Posterized”? Or just open a photo in Photoshop, and posterize it in two or three clicks.

It’s Posterized. And in Photoshop, you can adjust the level, i.e. convert the photo to five colors, or 20 colors, etc. Takes about 2 seconds.