Language is use
Based on that observation, we should immediately reject the idea that terms like burger or sausage should not be allowed to be applied to plant-based products. The only reason we associate them with meat, is that in the past they have always been made primarily or exclusively from meat. But that's a historical artifact. Their main identifiers, the way we classify something as a burger or sausage, is their shape and how we eat them.
The argument becomes a bit more complicated when we start talking about words we associate with specific parts of an animal, such as steak. Indeed, this is likely the same reason why terms like almond milk are banned: these categorise a specific product. Milk is not just any white liquid, it is produced by lactating animals. Of course almond milk is a dairy replacement, but that's not the same as saying it's plant-based dairy. So what are other reasons we should ban or allow these terms?
At this point, from a usability perspective, does it make sense to use the same label for both. Outside of a courtroom, words never have precise definitions that we take from dictionaries: meaning of words is shaped through how we use them. One important factor is what best helps us communicate, to make sense of the world and what other people say. If one product is the same for all practical purposes as the other, then we will generally use the same word, for no other reason than convenience. In fact, it goes so far that we can use specific brand names, to refer to generic products. We often don't talk about plastic containers, but about Tupperware. Okay, no other company can label their product as Tupperware, but that's an IP issue, not a language issue. Nobody is confused when you go to a store to buy Tupperware and you come home with a different brand.
Faux meats are meant to replace actual meats. To make it as easy as possible for people, using the same label, conveys better what the product is, than using some new word. And sure, vegan steaks are highly processed foods, but the issue is not health. You can label a bag of crisps as vegan, even if going through that bag in one sitting is not particularly healthy. For people who want to lower their meat intake, having a product labelled steak is convenient, because they immediately know what to replace their conventional meals with.
Vegan cheese and cheese-like
The ban on dairy milks thus seem justified from that perspective. I do, however, want to offer a counterpoint to all this: everybody calls it almond milk. I have literally never met anyone who asked for almond drink or something of the like. The meaning of language lies in its use, and it's very clear that almond milk is the term people use. Forcing companies to label their products differently is not going to stop that. The language of the law is just not the same as the language of human society. Legal language is a means of exercising power and control, and that is a very different ball game.
So why then do I disagree with the Guardian on terms like cheese-like and yoghurt-style? Simple: these labels clearly show that it is not actually cheese or yoghurt, but something that is made to give a similar sensory experience. Even if we agree that vegan cheese is wrong, because cheese has to made from dairy, cheese-like inherently means it's not cheese, and so there is no reason why it should not be allowed. For the same reason, IKEA can sell me artificial plants (which they have succesfully done; I cannot keep real plants alive), and I have yet to meet a lobby group that wants to ban this term. And of course, we have been using terms like peanut butter, and more generally nut butter, for so long, that the argument that these products need to contain dairy is plainly disindenguous.