Yes it has. Consider just one of the changes - the size of the new image. It's smaller, so to re-establish the correct viewing distance (which is fundamental to the whole concept) you must move closer. Therefore you can see more detail, and therefore the DoF changes.
When I say image, I use the classically accepted meaning for the word image - the one used by lens designers, Optical engineers, physicists, Opticians and the like - that is the image formed by a lens in a given plane (in this case on a film or sensor)
If the plane is unmoved, then no amount of changing the size of the film or the sensor will change the size of the image THE LENS produces. That is very basic optics
I look at a camera as 2 separate parts, a lens and the body.
This whole issue arises because for some reason, the photographic industry wanted to "normalise" a sensor size to a 35mm equivalence i.e 35mm = full frame (I guess because they built full sized looking cameras and slapped a DX sensor in them, they then realised they could release poor quality lenses that actually performed really well on the limited size DX sensor but didn't perform at all on a real 35mm format). For any of us that shot/shoot large or medium format, or for any of us that has used old 35mm lenses or MF lenses on a new modern digital camera, this was clearly just a anti-negative marketing ploy. The irony with all of this is that we argue about lens quality, when old (and new) MF/LF lenses need to produce a clean sharp image on a comparatively massive area compared to a DX lens
For me photography is where science meets art, and the science of optics is pretty simple, and did not need the marketing guys putting a rug underneath it. When you formally study optics (like I did), you start off with a simple example like: Here is a 10 dioptre lens. You can then form an image with the lens, and measure how far from the lens the image is, and how big it is la la la..... You can the progress to learn about things like COC's and aberrations, and best form, and asphericity, and you can still go and set the thing up on a optical bench, fire a beam of light through it and make real measurements. When you learn like this you very quickly disassociate the lens system form the measuring tools The measuring tools simply measure simple things like where the lens is, and how big the image is, and where was the image formed, how wide was the fringe etc..
What is totally annoying is when the marketing people and then the industry pundits take something that is actually very simple and black and white (like the image
is 2 MM high", and then push it through the mould of equivalence and come up with "well if it was on a fx sensor AND we then imagine they were printed at the same size, the image is actually 1.75mm high (or whatever)" when clearly it isn't
For me photography is more than shooting DX or FX digital, I shoot film, MF, the odd large format, pinholes, zone plates and pretty much everything else available to me. To pull this off, I rely on knowing facts, like 50mm = 50mm. The moment we add equivalence into the mix, I then have to skew everything to fit and almost calculate everything twice