No, they're not exactly the same.
"To learn" ("a învăța") means to acquire some new knowledge or skill, by whatever means or methods. The emphasis here is on the actual acquisition of that knowledge or skill, i.e. the person doing the learning is gaining something useful.
"To study" ("a studia") means to make some effort and use specific methods, with the intended purpose of learning (acquiring) some new knowledge or skill. The emphasis here is on the actual effort and/or methods used (usually reading, careful observation, etc.) in order to acquire that knowledge or skill.
Native Romanian speakers often confuse these two because of the pervasive usage of "a învăța" when referring to, for example, reading for some upcoming school test. But remember, just because you're studying doesn't necessarily mean you're actually learning something, or vice versa.
And for that reason we would be more likely to say "studies", or "is studying" here. For one, we don't generally use "learn" in the Present Simple, as in the answer here, and although we say that someone "is learning Romanian" (speaking generally), when it comes to school subjects or things like accounting, the verb "learn" is used more to talk about the acquisition of specific skills, eg:
"Tomorrow we will learn about double-entry bookkeeping"
Addendum 1 - We usually learn "about" something, or "how to do" something, but rarely "something", at least not in this context.
Addendum 2 - From Learn Romanian with Nico, conversations 17 (1):
"Nu am venit, pentru că azi am avut un examen important și am învățat toată noaptea."
We wouldn't use "learn", here, but "study", which is also what Nico gives in her translation.