There seem to be a number of posts on other forums (I know, get a life!) insisting that, for example, an M43 75/1.8 is equivalent to an FF 150mm 3.6 (or thereabouts) if such a thing existed, in all respects including exposure. Surely this is not right.
You are right to question this.
As far as depth of field, it is a good approximation, but there are conditions where this falls apart--usually with macro photography.
The problem with equivalency, as Matt points out, this is really just a collection of geometric relationships that require a lot of assumptions and do not take how photography actually works into account. F-number is lens speed, not depth of field. Yes, you can model the human visual system in its perception of a print (and depth of field is the area that is perceived to be sharp in a print and nothing inherent in the camera image). But f-number does not represent depth of field per se.
The other red herring is the idea of "light gathering." That argument is great if you want to pick winners and losers as the larger sensor always has the potential of intersecting more light. But here is the problem--pixels are discrete--what one pixel "gathers" has no baring on what another one does. Now, larger sensors can have larger pixels assuming the same pixel resolution (another assumption)), but the advantage is the pixel has a greater area, not that they cover a larger format. Pixel area is also known as detector size. But play a thought experiment: if pixel were perfect in that they would count every photon and have no noise, what would a larger sensor provide in and of itself?
The other problem with equivalency is that it assumes an absolute frame for comparison. But we don't use cameras the same way and there is a large variation in how a photograph can be taken. If you take random images and displayed them, could you actually identify the sensor size that produced the image. You would not probably succeed beyond random chance. This is why when equivalent images are compared, they need to be under very controlled conditions and then compared at extremes: 100% monitor view, high ISOs, extreme lighting, etc. None of those condition represent actual viewing conditions of a final image.
And lastly, equivalency is cherry picking. The comparison aligns the variables that supports an argument, but it is impossible to make every variable "equivalent." As a simple example, do you fix the number of pixels or the pixel pitch? The equivalence would argue to fix the number of pixels, but it is the pixel pitch (detector size) that is related to pixel efficiency. Their argument is the the image should "look like" each other. But here is the thing: if pixels are not resolved by the human visual systems, can a viewer distinguish between two images of different pixel resolutions? You don't have to have the same number of pixels to make the same visual response.
Yes, equivalency becomes an interesting learning tool to understand some of the relationships to sensor size and things like depth of field. But it is a model to make comparisons. It is not a scientific imaging theory.
Sensor size does not change focal length or f-number. Just like you cannot use focal length or f-number to change sensor size (I mean if they those are real relationships, then they work both ways).