Sure, you can. I have done it quite a few times: reading from one DB to write to another, or reading from two DBs to compare the data, etc.

Although, as far as I can remember, when I did it, it was always DBs of the same type (using two Oracle DBs, or two mySQL DBs, but not a mix), I cannot see any reason why it would not work with DBs of different brands or from different providers, at least not in general (there might be some very specific cases where it does not work smoothly, but I can't think of any such case).