Research administrator in the US here. PI rights are very confusing for researchers, especially new ones. Many think that "PI" = the person who had the original idea, which is not the case. The PI is the one who will be overseeing the research, whether it was their research or not. There are many cases where PIs will be "in name only" to facilitate others to carry on the bulk of the work. Some junior faculty use this approach with a senior PI to help get them their first grants.
In any case, the institutions grant PI rights according to their own policies. In US Universities, this is usually restricted to ladder/tenure-track faculty. As such, postdocs usually are granted PI rights on an exceptional basis, e.g., they have a senior faculty member backstopping them. In 13 years of this work, and several hundreds of proposals, I have never seen a graduate student as a PI unless they are a submitting a fellowship. Graduate students are trainees, and by definition unqualified to supervise research. Postdocs are given rights in an exceptional case as they are quasi-trainees and somewhat independent.
Check your institution's PI Rights policies to see who needs to be the PI -- it is likely limited to tenure-track/tenured faculty without additional paperwork required. Personally, I would never advise a postdoc to take on a PI role without a very good mentor behind them. The amount of administrative burden is more than they anticipate. Institutions throw a lot of resources at faculty to help them through this; this is not necessarily the case for postdocs.
As for the other part of your question; you should inform the student of this information as well. If you want to be part of the research community, it is best to be forthright about who has what responsibilities and not make decisions in secret with the hopes of acquiring some sort of prestige. You will soon find out that many senior faculty are all too happy to give away the job title of "PI" to someone else who is willing to do the paperwork associated with grants and contracts--which is usually what this title conveys, not that they are the sole person with the ideas and intellectual property.