We call it a "Pokeball". [Nice and straightforward.]

[Tiny Silvally looks perfectly comfortable. It plunks itself down and curls up a little, even.]

Another difference is that, where I'm from, you have to look hard to find someone who isn't carrying at least one. The Pokemon inside might be a pet, or sometimes a...worker [still not sure wtf to call horses!! nobody seems to know!!], but usually it's more like a partner to them.