More like dress pants, fancy shirt and polished shoes. Or for us females: skirt and pumps?!? Would be a little weird instructing mat exercises wearing a skirt and balance exercises in high heels...
Aside from clichee I always thought that tracksuit/trainers was the most practical for moving around lots and depending on your treatment (and your client's size), a treatment session can equal a good gym work out!

I think "professional" look differs between professions, not everyone has to look like a lawyer to portrait professional competence. I would still consider eg a builder to be competent who wears steel caps or a lab tech who wears a coat.