It is possible for a postdoc to be at a different university from the grant-holder (PI), and supervised virtually as a result. It probably requires an unusual source of funding, and entails either frequent visits or else a co-supervisor/collaborator based at the postdoc's institution.
- An example
Here is an old job advert from the CGSP/CPSM, a Canadian research project in social policy. One can find by a little googling that one of the appointees is based on another continent, where they are co-supervised by a professor who is not listed as a collaborator or partner in the CGSP/CPSM. So their supervision by the PI would be mostly virtual (apart from one workshop planned as part of the programme).
In the job advert, you can see some very careful wording about supervisors and co-supervisors to permit exactly this to happen.
- Something similar in mathematics?
In fact I know the PI in the example above (that's how I heard about it at all). She told me that those postdocs were in some measure based on the Fields Institute postdocs in mathematics. These are not precisely what is being asked for in the question, but show some similarities.
Academics apply to organise "thematic programs" at the Fields Institute in Toronto, and the institute hires postdocs to work on those programs. Unlike the scenario in the question, the postdocs do physically come to Fields. However the organisers could be anywhere in the world, so their supervision of the postdocs may be largely virtual. The postdocs also aim to work with faculty at "sponsor universities" of the Fields Institute and many of these are physically quite distant from Toronto.
The supervision is not entirely virtual as there are workshops and lecture courses every month or two to bring people together physically.
Of course it can happen that a PI or postdoc has an extended visit at another university or institute, or that the PI moves and it's not possible for the postdoc to follow. If one wanted to see the different ways a "virtual postdoc" could go in practise then those cases might be the place to start.