I will answer your two “why”(s).
To your first “why” : why do they run their restaurants in Europe under the name of “Chinese restaurant”. Clarity is, there are some restaurants like that, but not common enough to find it in anywhere of Europe, for if you offer me a mission to hunt down this type of “Chinese restaurants” in Europe, it spend an amount of efforts, maybe I could easily miss it even if it was just locally right next to me, but if you ask me to find you any Vietnamese restaurants that are named as Vietnamese cuisines, I can easily find you any.
So why are there Vietnamese restaurants that are named as “Chinese restaurants”? For the sake of business, you need to attract people via utilizing their ignorance; it has been the case that the word “China” refers to not only 中國but an ambiguous giant region in the Far East, this was NOT a dictionary usage, but it was how people make assumption on the terminology, and with that assumption, any things which have to do with Mongolic Asian (East Asian and south East Asian) , are grouped under the labels of “China” and “Chinese”. So, it has been the case for people to associate the word “Chinese/China” with the foods(and other cultural elements)that are originated from the Far East. In reality, it is quite a common to find an Asian restaurant in the West(not only Europe) , which names itself as Thai restaurant, Vietnamese cuisine, and even particularly “ Phnom Penh beef noodle soup”, in other word, your “why” is relied on an inaccurate bias.
To the second “why” : they don’t pretend to be Chinese , your “why” is invalid. If the restaurant naming is a sign of identity pretending, then it was so easy for me to find some Chinese who pretends to be Vietnamese. I’ve come across some of these restaurants in at least Vancouver and Toronto in Canada, which are though operated by Chinese but yet selling Pho, Vietnamese coffee and Gỏi cuốn etc. these Chinese are not even the Cantonese from Saigon, they don’t speak a single word of Vietnamese. If you want to deny my claim of experience, so can I do the same toward your European claim. The funny one I encountered in about 2 years ago, in Richmond BC Canada, there’s a “越南牛肉粉” shop , all employees staff are mandarin speakers with the up north accent, the “Er” sound was way deep and thus unavoidable; maybe I would considers a chance that their kitchen is still occupied by Cantonese from Vietnam, or the boss behind the scene is a Cantonese from Saigon. The point is, I wouldn’t be silly enough to say that these Chinese are pretending to be Kinh people.
And about restaurants naming, here’s another scenario to explain it, maybe it doesn’t happen in Europe yet, but it happens often in California, something like this:
a mainland Chinese youth who’s second generation rich (富二代), on either a green card or F1 visa, driving fancy cars and cashing things without a manner , he also bought a famous “Chinese restaurant”(Mongolian BBQ+ WOK) from someone, due to his dad’s command(you got to do something !) , but he is too dumb and couldn’t maintain it, eventually , a Vietnamese business man bought it from him with an offering of extremely low amount. This Vietnamese man is now keeping a famous restaurant in town, and he also decide to keep the original naming, for the sake of it’s reputation, and there are new items on the food menu, which are Pho and Banh Min.