I read the specs and reviews and I can’t see how this could be hazardous. The thing I would be worried about is if there’s any coating to burn off during the first use.
So I guess I would put it together and use it a time or two before I have my birds near it.
I would air out the rooms well after the first use.
I’m assuming the composite materials are safe to heat because it’s approved for household use. But I do see the materials are not real stone so I have some concern about their heating.