“Costume Shop” has long been the most popular name for these spaces in the US, as the name seems to be a direct co-equal with “Scene Shop” the most popular name for the Scenic & Prop building areas in these institutions. Positively it also has implications of (mostly male) skilled trade work relating to “Machine Shop”, “Auto Shop”, and other skilled trades that generally pay better than garment (ie “womens”) work. “Costume Shop” is also generally perceived as a less pretentious name by most workers. Negatives include being confused with costume sale and rental stores, and getting phone calls intended for them, and a general confusion for non-theatre people when reviewing resumes of former workers.
“Costume Studio” is increasingly used in academic theatre because of its association with Art Department studios, which in academic hierarchy places it higher in the intellectual/artistic status, rather than the Wood Shop, Auto Shop, trade school/adult ed status which is lower. This is assumed to help with the tenure/promotion of the faculty managing the area, and possibly funding and respect within the larger institution. It is positively associated with artistic exploration and experimentation, but sometimes negatively perceived as more pretentious by student workers, and is less usual by far. Advantages include less confusion with commercial costume sale and rental stores, and more clarity on student worker resumes.
Both of these terms are good, and understood within the field to be essentially the same thing, as a costume shop/studio combines the technical workshop and artistic studio in one space, doing both types of functions in tandem. The choice of which is best to name your space depends largely on the environment in which your shop/studio lives. I myself (Tara Maginnis) called the space at The University of Alaska Fairbanks the “Costume Shop” for my 18 years there, as Alaska in general has a strong bias in favor of “manly” trades and pursuits, and skilled workers in these trades often pull in salaries and respect higher than intellectual workers in academia and office work. When I moved to Diablo Valley College in Pleasant Hill California, I promptly renamed the former “Costume Shop” there the “Costume Studio”, because the SF Bay Area’s most famed status workers are those who do Art for Pixar and Lucasfilm, write code for Google, and the ultimate wealthy person vacation is to help pay for and work with someplace like the art studio The Crucible in Oakland to make a sculptural art event at Burning Man.
The choice of which term to use should be focused on which one will help your workers and department and yourself most in terms of pay, resume, funding and social status in your community.