I just... don't understand this mindset at all. To all intents and purposes, egg is an entirely utilitarian ingredient in mayonnaise, as an emulsifier for the oil and water. In the past, this would be our only emulsifier readily available for such a dish, and now we have many more alternatives. This product (supposedly, I don't live in the US so have never tried it) tastes and feels exactly like a good mayonnaise, and shares the two largest parts of the ingredients, why shouldn't it be referred to as a mayonnaise?
I think the American "cheese" thing is a bit of red herring too. What people are actually referring to is the fact that it's actually a cheap, heavily processed product that doesn't really bare any resemblance to cheese as we have known it before. This complaint about American cheeses is not usually dished out to, say, Treeline's nut cheese brand — presumably because it looks like "real cheese".
It seems incredibly arbitrary to just pick one ingredient and say "hey, this is what makes this a real version of [x]", especially when that ingredient is apparently so unimportant to the actual consistency and flavour of the product. It's as if we'd arbitrarily decided that something would stop being mayo if the type of oil wasn't canola or something.