The ordinary English idioms are:
One studies under a master artist or craftsman; it’s a metaphor drawn from the old guild system in arts and scholarship, where an apprentice or student put himself under the master’s discipline.
One studies in or at the studio which the master directs.
COCA gives no examples of studying under a studio. Google yields 105 instances of under the studio of, mostly referring to artists artists and productions “under” a film, recording or animation studio. Eliminating duplicates leaves 21 unique uses. Of these:
- Four occur in student biographies of pre-20th-century painters
- One occurs in the biography of a floristry instructor
- One occurs in the resume of a high-school dancer
- One occurs in the self-written biography of a Russian jazz musician
- 14 occur in program-style biographies of musicians, all of East Asian origin.
I have no hesitation in pronouncing “under the studio of” a mistake by persons unfamiliar with the accepted idiom.