Who owns the research data at US research institutions?
No one.

The short answer is that “data” as the term is understood by US law is not covered by copyright, thus in popular parlance it is not “owned” by anyone. If you google “can data be copyrighted” you will find various articles explaining this in more detail. Here is one.

The longer answer is that, as seems to often be the case in matters of law, the legal meaning of “data” seems to differ slightly from how people use this word in normal speech. What academics might refer to as “research data” may also include aspects of what is referred to in some sources as “presentation” or “compilation” of data, and to the extent that this is the case, copyright might apply to it after all. In that case, the copyright would be owned by some combination of the PI, graduate students, the institution that employs them, and any funding agencies who might have funded some of the research, depending on the policies of the institution and funding agencies. For example, at my own university, researchers own the copyright to the work products they create.

I also notice you are mixing up copyright and patent rights by mentioning them together as if they were the same thing. They are not. Data is not a form of invention so I don’t see how patent rights would apply to it. But more generally, if a graduate student and PI come up with any patentable inventions, those would be subject to quite different rules and policies than those covering copyright. For example, at my own university, patent rights for researchers’ inventions will generally belong to the university, though the inventors can receive a share of any royalties generated from their inventions. As far as I know this applies equally to graduate students and regular faculty. But you should check the policy at your own institution.

Disclaimer: I am not a lawyer.