Based upon what, precisely? He's the Vicar of Christ on earth, meaning the visible head of the Church. And since there is only one Church, there can only be one Vicar. To say he can choose to be the head of multiple churches implies there is more than one church with Christ as its head.
Edit: Here's Session 4, chapter 3, section 3 of Vatican I on the Supreme Pontiff:
What I meant was, there is nothing to stop him from being head of multiple organizations and the Church at the same time. Certainly what he is doing, to say the least, is altogether wrong, but aside from his own conscience, he can be head of whatever he wants.
ETA: Fr. Hesse, in one of his talks (which I will try to find later) briefly explains it.