Directly answering the question, no. Unit focusing doesn't inherently decrease the depth of field (DoF) compared to element focusing. However, there are lots of hidden or implied possible variables. But based on the wording in your question, there is some nuance that needs to be made explicit.
[Moving the objective relative to the film plane to set the focus] also effectively increases the magnification when focussing close-up, because the image circle at the distance of the film plane gets larger as the objective is moved further away. This also requires exposure compensation (sometimes called ‘bellows extension factor’).
I wouldn't word it that way. Yes, it makes sense to talk about magnification when in the close-focus regime, as opposed to talking about subject-focus distance. But there is a 1:1 (ableit nonlinear) mapping between object focus distance do and magnification M, as defined by the thin lens formula (1/f = 1/di + 1/do) and the definition of magnification, M = hi/ho = di/do = f/(do – f) = (di – f)/f .
The "bellows factor" exposure compensation is required for close-up focusing, whether using an actual bellows, or internally focusing (as you called, element focusing) lens. This is because the exit pupil of the lens is significantly far enough away from the image plane that the small numerical aperture (from the image plane's point of view) is dominant enough to become material in the exposure calculation.
With element focussing ... It’s my understanding that there is typically some small effect on magnification, but that on a typical general-purpose lens (setting macro lenses and some other special cases aside), designers try to minimize this effect. No exposure compensation is required.
Setting aside that magnification and focus distance are directly, formulaically, tied together, you are hitting on one thing that is true, that you didn't state: internally-focused lenses don't necessarily have constant focal lengths throughout their focusing range. This is especially true on some zoom lenses. But even unit-focused lenses, with a fixed-focal length, do not maintain a constant angle of view throughout their focusing range. This is known as focus breathing. In fact, only specifically-controlled internally-focused lenses eliminate focus breathing by changing focal length inversely to the would-be narrowing angle of view as the lens is focused closer, keeping the actual observed angle of view constant.
Does unit focussing also decrease depth of field compared to element focussing due to the higher magnification/effectively longer focal length?
(emphasis mine) This is the part of your question that I think is backwards: the focal length of a unit-focused lens doesn't change. The focal length is a physical property of the lens system, determined by the shapes of the lens elements and their relative distances from each other. Because their shapes don't change, and the relative distances of the elements don't change, the focal length f of the unit-focused lens is absolutely invariant.
Conversely (and contradictorally), loosely speaking, you can consider the narrower angle of view of a close-focused unit focused lens an apparent increase in the lens's focal length (as compared to the lens's measured angle of view at infinity focus). In this sense, an internally/element-focused lens controlled for focus breathing will have a shorter real focal length when close focused, as opposed to focused at infinity. Thus, the DoF calculation would have to use the close-focus adjusted focal length, as opposed to constant f (in the case of a unit-focused lens).
I.e. if I have two lenses of essentially the same design and focal length, focussing on an object the same distance away, but I focus one as a unit and one using the individual elements, will the former have a smaller depth of field than the latter?
We're going to have to be a bit more precise. Assuming that a particular internally-focused lens's focal length is kept constant throughout its focal range, then if we focus that lens at infinity, then mount it on a bellows focusing rack onto a camera, and comparing to the same exact lens on a camera without a bellows (i.e., relying on the lens's internal focusing), then at a given focus distance, no: the unit (bellows) focused lens will have an apparent longer focal length, and therefore deeper DoF, than the internally (element) focused lens, with the constant (i.e., marginally wider) angle of view (and correspondingly marginally shorter focal length).