Since I just moved to a new city, I'm in the process of shopping for a new Pilates studio. What I've found is that even among studios that teach classical Pilates, there can be a bit of variation in terms of teaching style, and how they might want you to do the exercises. For example, some of the verbal cues they use to get you to engage your core may be different, but it all translates to roughly the same thing. Within a studio it is reasonable to expect that there is generally some consistency in that regard.

I think in my case there is also a "getting to know you phase" when going to work out in a new environment, since I'm not starting from scratch.