As far as ordinary language is concerned, the basic alternative at issue is whether or not we want to apply food categories like "meat" or "mayo" or "milk" by reference to an underlying substance (such as animal flesh or egg yolk emulsions or mammalian secretions) or by reference to the category's characteristic function. If what we mean by "mayo" is "white, tangy, oily stuff that you spread on sandwiches or make tuna salad with" then vegan mayonnaise is possible. But vegan status precludes the possibility of mayonnaise-hood if what we mean by "mayo" is an emulsion made from eggs and oil. The same basic alternative applies in the case of nut milks. If by "milk" we mean "a whitish aqueous emlusion of fat, sugar, and protein that you might put on cereal or drink a glass of or put in your smoothie," then sure... nut milks are possible. But if milk refers to a substance secreted by hairy mammalian teats, then nut milk ain't milk. Neither is soy milk. As Lewis Black puts that argument, "We know there's no soy milk because there's no soy titty." I'm not convinced that's the best way to view things, and am content to make, buy, and use nut and soy "milks" without any sort of linguistic or conceptual shame. It's not like anyone's confused about whether or not these "milks" are real "moo-cow **** milk," as Mr. Black calls it.
These distinctions are mostly without a difference in our everyday lives, but they can come to matter in the context of our byzantine regulatory environment. Is mayo essentially an egg-based product? How the law decides could have a multi-million dollar impact. I seem to recall Hellman's trying to use the regulatory baton to badger the makers of "Just Mayo" into changing the name of their product. A lawsuit to burden their much smaller competitor. Kind of a dick move on Hellman's part. But so is putting a picture of a freaking egg in the logo of your egg-free "mayo." (It appears that the makers of Just Mayo have since removed the egg from their logo.)
Are hot dogs sandwiches? What about burritos? There are legal rulings on these matters that impact real people's bottom lines. Are tomatoes fruits? Yes, botanically -- they're edible ovaries. But for customs purposes, the feds say they're vegetables. Are the mutants in the fictitious X-Men universe human or are they non-human? Toy Biz, Inc. v USA argued that their X-Men figurines should not be taxed as dolls, but as toys on the grounds that the X-men are "non-human creatures." What's the metaphysical truth about the humanoid status of the X-Men? Are they even men at all? The world may never know. But for the purposes of import tariffs, the X-Men aren't human. But I digress...
