Astral projection is a mode of travel, period. Destination, however, has made it a game of make believe, with all sorts of magical fake creatures.
Now, about the phoenix, no. You cannot have one as a pet. They don't exist, accept as myth creatures who some people believe to be a creature of destruction, or rebirth, as the story is they reincarnate from their own burnt ashes. So you see why you can't own one.