The confusion is in your first sentence, where you identify a gravitational force as "being centrifugal" or "being centripetal". That's the basic error, which then sets you up for "but that's fictitious, isn't it..."
Let's fix that error quickly and simply.
Classically, the sun attracts the earth, and the earth attracts the sun, due to their mass, via gravity. (I'll ignore general relativity and warped spacetime and such, but be aware that's a more sophisticated and apparently correct way to see it).
But neither of those forces should be defined as being centrifugal or centripetal without a bit more thought.
The suns pull on the earth causes the earth to accelerate. The earths acceleration results in an elliptical (almost circular) orbit. That's a genuine force. If it was a physics question we would identify it as a centripetal force - a force pulling toward a fixed centre, or fixed point.
Similarly the earths pull on the sun, causes the sun to rotate round the earth. That's also acting as a centripetal force.
Technically both rotate around a point called their baricentre, a sort of "centre of mass and gravity" for the 2 objects. Each is causing the other to rotate around it. However, as the sun is so much bigger, we only see one object rotating, but in reality both are circling.
But there isn't a centrifugal force in the system. Not really.