The mathematics and geometry your working with apply to SIMPLE LENSES. A single lens element with a simple convex curvature of negligible thickness. They also technically only apply at hyperfocal distance in air (vs. water, or vacuum, etc.) The Wikpedia states this:

As mentioned above, a positive or converging lens in air will focus a collimated beam travelling along the lens axis to a spot (known as the focal point) at a distance f from the lens.

And:

If the distances from the object to the lens and from the lens to the image are S1 and S2 respectively, for a lens of negligible thickness, in air,

That's a long list of limitations. In those SPECIFIC circumstances, a simple, thin lens in air will perform according to the formula specified.

As your focal plane moves closer to the lens, even a simple lens is going to behave differently than the ideal mathematics for a simple, thin lens in air would indicate. There are a multitude of problems involved in the use of simple lenses, in addition to the magnification factors your asking about there are also numerous optical aberrations that occur making simple lenses less than optimal.

A real camera lens is much more complex, involving multiple lens elements that bend light in such a way as to maintain not only a flat field and the sharpest possible focus from corner to corner, but they also maintain the proper magnification ratio, entrance pupil size, etc.

I've used both 400mm and 600mm lenses. The expected change in subject magnification between those two is 2.25x. At 600mm, the subject should be 2.25x larger than at 400mm. In practice, that is generally the case. There are small variances...for example the Canon 100-400mm lens actually tops out at around 390mm instead of actually being 400mm, and depending on your exact focal plane, there can be other shifts. In general, though, my experience is that with the EF 600mm f/4 L II lens, subjects are indeed about 2.25x larger than with the EF 100-400mm lens.

Complex multi-element lens designs don't have the same geometric effect on light as simple lenses do. There are a multitude of factors that go into choosing what elements to include in a camera lens, not the least of which is choosing a front element design that is large enough and curved enough to gather all the necessary light from the right angles to support the specified FoV. Additionally, the front element and all the elements before the diaphragm must produce the necessary entrance pupil magnification to actually achieve the specified F-Ratio. All the elements in concert, including floating elements that move with focus or with changes in zoom, must work together to produce the required subject magnification.

Sometimes compromises must be made in order to achieve good IQ, or to achieve the desired zoom range, etc. which in turn often result in less than ideal behavior (even at hyperfocal distance). High quality lenses will usually be near to ideal, however lower end lenses may not be ideal. Changing focus may change focal length, changing zoom may affect focus (i.e. the lens is not parfocal), etc. Regardless, the geometry and math involved in complex lenses is likewise more complex than with a simple lens. The results can be similar, however they are not necessarily the same. For demonstration purposes, we often assume that a complex lens behaves like a simple lens...at hyperfocal distance. Even at hyperfocal distance, while they will perform largely like a simple lens, they will rarely perform identically. At closer focus distances, they can perform quite differently than a simple lens, depending on the actual design.