Zooming in with the lens will also change the background because by zooming in your are changing the focal length. Also there is the lens distortion to consider as well. A portrait taken at 28mm will look very different than one taken at 150mm no matter what you do.
How does the focal length affect the background? Are you referring to the focal length's relationship to the depth of field? Because otherwise, I guess that the effect you're referring to ("A portrait taken at 28mm will look very different than one taken at 150mm no matter what you do.") is an artifact of the same thing the GP comment talked about, which is the effect of the distance between the camera and the subject.
Wide vs. tele foreground/background lens distortion, not just the field of view (framing). Some call this choice of lens "lensing" and as noted, the distortion of a 28mm wide vs 200mm tele are part of their physical design, optical "laws" being what they are.
GP had it exactly right, perspective distortion ONLY depends on the distance between objects to the camera (entrance pupil). The focal length literally does not matter at all.
(This is an interesting point of contention, because conventional wisdom up until a few years ago was exactly what you said, and relaying what GP said got you tarred and feathered as an imbecile in the usual "photog forums". It's also one of these things which were never in doubt at all, and are very easy to check for yourself.)
