Perhaps you are
I think it's SO. Maybe LA? (Reasoning ahead)
Some call me unemotional and austere
DO perhaps? I thought you may be referring to just a normal deer (a doe), but both John Doe and dough also came to mind.
but I will bring the sunshine into anyone's life.
Clearly, this must be ray for the note RE.
I am my next note, though we are not the same.
ME. Referring to itself. The notes ME and RE are not the same, though perhaps similar.
I'm a common complaint from students of language.
(It's all ----- to me!) Maybe it's ”So far.” The note FA can sound like far with an accent. People complain about distance all the time.
Finally we have:
After the note FA we would have the answer SO. However, might be a stretch, if we count ”so” in ”so far” then it may be that the next note is LA.