It seems to me that you get thrown off a little by contractions of the compound tenses in "not only"[....]"but also" [....] sentences. The verb in your "not only " phrase is does study, which is the present emphatic tense. The present emphatic tense is a compound tense using the auxiliary verb "do" or "does". When you make a compound tense negative, it is more natural to put the auxiliary verb before the "not", but it is still part of the whole compound verb -- in this case, "does study." The present emphatic and the simple present have basically the same meaning. You could have used the simple present in the first part of the sentence:
He not only plays the piano well, but also studies well.