The question is in the title. Let me clarify what I mean by « biological AI » : it looks like an oxymoron if you understand biological as natural and artificial as non-natural, but let understand biological as organic and AI as what it actually does (and not as what it is made of). On the other hand, I am (obviously) not asking whether the human brain was designed...  
AI is here understood in its current classical sense, and not in what it could be in the future with (for example) quantum computers, so that if one can shows (for example) that the (humain) brain involves some quantum processes which cannot be simulated classically, then the answer should be no. Note that the quantum processes were just used here as an example, the question does not reduce to the existence of such processes.  

*Reformulation*: Is there (experimental) evidence that the (human) brain is “essentially” more than just AI?