In simple terms, if we fix the subject magnification by adjusting the distance between the camera to the subject, then the size of a blur circle for a background object at infinity is going to be proportional to the entrance pupil diameter, regardless of the actual f-number or focal length.
For example, if you take a 50mm f/1.0 lens, and a 600mm f/4 lens, the entrance pupil diameters at infinity focus are going to be 50mm and 150mm, respectively. Taking these as approximations to the entrance pupil diameters when both lenses are focused so that the image magnification are both at 0.01 (i.e., in each case the subject is located at a distance such that one meter of subject height will correspond to an image height of 1 cm), then for objects behind the subject, infinitely far away, the blur circle from the 600/4 lens will be approximately three times the diameter of the circle produced by the 50/1.0.
However, this scenario is not applicable when background objects are not infinitely far away, which is almost always the case. The reason is that when objects not in the plane of focus are relatively near that plane of focus, the effect of f-number on blur circle diameter is stronger. Therefore, a lens like the 85/1.2L will exhibit a more rapid increase in the blur circle diameter as a function of background separation from the subject, but achieves less maximum blur than a slower but longer focal length lens like the 300/2.8. Again, the comparison we are speaking of is for a constant subject magnification.
This is why very fast aperture lenses of modest focal length have a distinctive look: although you could get more background blur with a telephoto lens, the combination of a relatively short focal length and a very fast aperture results in an image where you can see more of the background (field of view is larger), but the drop-off in sharpness is more dramatic. More background blur alone does not create this look.