Yes of course he is a doctor there. Unless a doctor is in a specialty that is hospital-specific (no reason for a separate office & private practice), he is not likely to be employed by a hospital. For example, I imagine that emergency medicine and hospitalist doctors are hospital employees but IDK who else.
Not all ER docs are employed by the hospital. I worked for one that contracted with the hospital to run their ER. The public never knew.
Yeah nobody works for anybody directly any more, it seems like!