Where horse lives is called?

Generally speaking, the building where a horse is kept is called a stable. This is the most accurate term.

Inside the stable, each horse has its own compartment. This is known as a stall or a horse box.

When they are outdoors in a grassy, fenced area, that is called a paddock or a pasture. You might also hear the word barn. But a barn is a more general farm building which can house stables, hay, and other things.

So, 'stable' is the specific word you are looking for. Hope this helps clear things up