I made continuous/discrete distinction more in order to take a jab at people that don't know measure theory and therefore think these approaches can't be unifie

I made continuous/discrete distinction more in order to take a jab at people that don't know measure theory and therefore think these approaches can't be unified. (Though I do know for the record that in some cases, like the ones you mention, there is no overarching, unifying theory.) Other than that I agree with you with everything up to the point where you say "Without any further context...". The dichotomy that you describe needs graduate-level mathematics to be properly understood in the first place. I'm not sure why (luck?) it seems you are biased by being surrounded by people who are competent at ML and math. I guarantee you that is not the case. If you review for any of the big conferences (e.g. NeurIPS) you will see that really fast. :(