I would argue that the owl is the same across all games, and that he is in fact a mystical being capable of both bearing the souls of others and traversing boundaries between worlds. That includes the boundary between the dream world and the real, as well as boundaries between alternate dimensions, i.e. between Hyrule and Termina. In LA he acts as a representation of the spirit of the Wind Fish, and in OoT he acts as a representation of Rauru. In MM, I would say that he is acting as a representation of the Happy Mask Salesman, as they are the only two characters unaffected by Link's time travel.