Creeping americanisms I guess - but we steal from any language! I'd expect a 'dance school' (BR) to have a 'studio' (somewhere with a suitable floor snd mirrors + sound, video, etc), but we also have 'studios' nearby without 'dance schools' (legacy 'community' project to anyone with money to hire). 'Dance school' (other) might refer to a (childrens') dance group meeting in a community hall - but sometimes with portable mirrors, ballet bars, etc.
It's off-topic for this thread, but the above subject fascinates me:
Here, I have danced at regular milongas at eight places in 2025: 5 are "milonga of a school which is also a studio dedicated almost exclusively to tango," 2 are "milonga of a school which rents the same non-tango studio for all of their milongas and classes," 1 is "milonga of a school which is also a studio, but which they regularly sublet to non-tango people," and 0 are "milonga not associated with a school or a dance studio." I have also danced at some balls and galas, which are always at big, fancy halls – the kind with marble floors, elaborate lighting systems, bar service, staffed cloakrooms, etc. – intended for, well, balls and galas. I've never run into the idea of tango at a community or multipurpose center, church hall, or whatever.
To first order, where I live, people normally assume milonga = school = studio, and if the equation doesn't hold, it's automatically a special event. Non-school and non-studio milongas are balls or something, and not regular. Schools without studios, moving from place to place, don't exist. Studios without schools do exist, but aren't used for tango. Generally, if you go out to dance at a regular milonga, you'll be going to the same place every week for years, and that place will have mirrors, barres, etc. and tango lessons at least some of the time.