You can only start with your initial guardian Pokemon. Pokemon of other types are permitted, most people have taken it among themselves to forcefully strip down their character and only allow them Pokemon of their own type. I rather like that idea actually, as I never considered it before. That would make for a rather interesting turn and I will totally mention that when the discussion thread goes up. The fairy would be an interesting concept, to make all the children but fairy struggle.
But yes sorry for not informing you sooner, I must've missed that point in your SU but only your starting guardian is allows on your team at the moment. And no, other Pokemon that you meet are not implied to be guardians just other Pokemon you've befriended.
//sorry if this seems oddly typed, I'm at a videogams tourney rn and I'm typing everything from a cell phone.