1. Toy horse: This is the most common meaning. A hobby horse is a toy, often made of wood or plastic, that children ride and pretend to be on a real horse.
2. Stick horse: A simpler version of the toy horse, usually consisting of a stick with a horse's head attached.
3. Historical term: In the past, "hobby horse" might have been used to refer to a small, sturdy horse suitable for riding by children or for light work.
There is no recognized horse breed specifically called "hobby horse." So, to answer your question directly, there's no breed of horse with that name. However, the term still exists and holds different meanings depending on the context.