While I understand your point that nut milk chemically isn't at all related to animal milk, I don't think that's the association that comes to mind when using a product. The association is in it's use. By calling nut or oat milk, 'milk', you're making it clear to the consumer what it should be / can be used for, which is a substitute for animal milk.

By removing the word milk, all you have achieved is confusing the consumer as to what it can be used for. Can I put it on cereal, is it safe to cook with, can it go in coffee, etc, etc.

There isn't a consumer alive who thinks almond milk came from an animal. This is obviously nothing more than industry lobbying gone wild.