Almost certainly not. That said, I think most of the problems could be fixed with three rules:...snip...snip...snip...
Here's the thing though. I don't think nut milk producers are trying to fake people out. A huge part of their value proposition is that their milk doesn't come from animals, just like goat milk suppliers aren't going to want you to miss that the milk comes from goats, not cows.
And they fall under that first rule. As long as the word "almond" comes right before "milk" and isn't in tiny print or some other nonsense, it should be fine.
Same for veggie meats and sausages. If their labeling doesn't make this clear, they will fire their marketing departments in an eye blink.
Unfortunately, that's often not the case, and it is likely intentional. The sorts of folks who make veggie burgers tend to be vegan, and there's a very definite subgroup of vegans whose not-at-all-secret goal is to get fewer people eating meat. So if they can fool people into thinking that they are buying meat without breaking the law, they will absolutely do it.
This is not true to the same extent for almond milk and coconut milk, both of which came into existence hundreds of years ago as localized substitutes for cow milk for cooking purposes in areas that had lots of coconuts or almonds and not a lot of dairy cows. They are highly popular for cooking specific kinds of foods that call for them, and they are both also popular among people who are lactose intolerant. Their relatively recent popularity among vegans is therefore almost *entirely* an afterthought, rather than the main reason that those products are made and sold, so you would not expect them to be marketed in a way that is misleading.
This leads me to conclude this has nothing at all to do with making sure consumers are adequately informed. That's a non-problem which, if it comes up, will solve itself. This is entirely about traditional producers wanting to hobble their competition. If I have a choice between "milk" and "pureed, pressed, and filtered cashews", which one do you think sounds more appealing?
I agree. This law is a stupid law, IMO. That's not to say that there isn't a need for a law, just that this one isn't the right one.
Anyone proposing these naming restrictions better come armed with charts and graphs showing there's a substantial number of consumers who, more than once, bought the wrong product and suffered a harm greater than mild irritation at their own carelessness.
The point is not that the harm is huge. It's that people were misled into buying something that they did not intend to buy, and that distorts the free market. That's literally why we have false advertising laws. It really doesn't matter whether we're talking about a $3 container of milk or a $10,000 custom motor scooter. Truth in advertising is a hard requirement for a functioning free market, and when companies play fast and loose with things in ways that actually mislead consumers, that's a bad thing.
So requiring that they be crystal clear that their products are not made of meat is reasonable. Requiring them to avoid certain terms is not, IMO.