Aside from the meaning of "dichotomy" as the phase of the moon we call "half moon" and as a technical term of art in biology meaning a bifurcation, the original OED gives two definitions of the word. The first is any two-fold partition (i.e., there's one side, the other, and nothing in between) and the second is a two-fold division. As an example of the latter, the OED cites a source that contrasts the "popular" theological dichotomy of man into body and soul with the Christian trichotomy of body, soul, and spirit.
The supplement adds to the second definition "something paradoxical or ambivalent," and gives these examples:
By a dichotomy familiar to us all, a woman requires her own baby
to be perfectly normal, and at the same time superior to all other
Their uncritical use of the 'Communist' versus 'free
The latter example serves to illustrate some of the sense of the charge of "false dichotomy," when you're asked to make a choice between two things that are presented (wrongly) as the only possible choices.
But if your knife is sharp enough to cleave a domain into two parts, then it makes sense that the parts be distinct in the criterion of the cut. Otherwise, you'd have no basis for making the incision in the first place.