Can someone collaborate between universities for a PhD?
It is not uncommon to have a formal PhD position at university X, a supervisor at university X, and a co-supervisor at university / institution Y. This may happen for example, if the PhD topic is on the interface of two subjects, or if Y provides essential experimental / lab / other facilities for the research.

It is also not uncommon to have a PhD at university X, a supervisor at university Y, and a co-supervisor at university X. This usually happens when a professor from X moves to Y some time before, or shortly after, the PhD is started. The role of a co-supervisor may be solely to observe the process and make sure that the research will succeed and the thesis will be submitted in time and in good order, to university X's satisfaction.

In the UK, the so-called Doctoral Training Centers, are sometimes organised between several universities, sharing the same direction of research. I have not heard of any examples, but I assume that a collaboration of a PhD candidate with several professors at different universities in this case is more than likely to happen. 

Another example, which can happen very naturally, is when a PhD candidate is closely involved in a research project, which is already collaborative.