Is it meaningful to ask whether the whole universe is rotating?
I think so. The Earth rotates, the Sun rotates, the galaxy rotates. Why not ask whether the universe rotates?
Is it meaningful in the sense of falsificable to ask whether the whole universe (including everything known/observable: cosmic background radiation etc ..., excluding everything not directly observable like dark energy) is rotating?
It's meaningful to ask the question. As to whether it's falsifiable I don't know. But you never know. I think this article was quite interesting myself: Galaxy sized twist in time pulls violating particles back into line . You can read things like this: "“The asymmetric Kerr metric as a source of CP violation”) suggests that researchers have neglected the significant impact of the rotation of our Galaxy on the pattern of how sub atomic particles breakdown ". Of course, that concerns rotation of the galaxy rather than the universe, and it is just a hypothesis, but it's interesting nevertheless.
Naively you would say, sure you can measure centrifugal forces. But centrifugal fictitious forces are defined with respect to a background in Newton's mechanics or indirectly as geometry in Relativity.
And if the background is somehow rotating?
Somehow I did get from Newton's bucket to headlines like "the universe is spinning". I am confused. E.g. in "Shape Dynamics" (Although the question has nothing to do with Shape Dynamics!) as far as I understood the rotation of the universe as a whole is dismissed as unobservable. Please correct me if I am wrong! Note again: Shape Dynamics has nothing to do with this question.
I'm not a fan of Newton's bucket or Mach's principle. And I am definitely not a fan of shape dynamics. But I am a fan of asking interesting questions, and pondering about the universe. Does it spin? Probably not, but it's meaningful to ask the question anyway.
Edit in response to revised question title:
Does a similar concept like centrifugal force exist for the whole universe?
I would say not. When you're in a car going round a tight curve you're subject to centrifugal force pushing you outwards. Ditto when you're riding a roundabout. But when you're in orbit, you aren't. You don't feel a force pushing you outwards. You don't feel any force at all, even though you're going round the Earth. IMHO you should think in similar terms for the universe. Whilst galaxies are moving apart, the "raisin cake" expansion doesn't resemble the motion that would result from some kind of centrifugal force.