Ultimately all metalwork needs to be electrically connected together, to ensure it's at the same potential. The safety/protective "earth" on the mains circuit needs to be connected to the car's chassis to ensure that the two can't float at different voltages, which could present a shock hazard.
You are not wrong, but I want to stir things up a little bit:
Note that what you describe is
exactly the mechanism which makes it dangerous in the first place.
Floating is of course safe by definition - there is no path for the current. You need to literally get between the two "live" wires (L&N, which are not floating respective to each other), almost impossible to happen by accident as a consumer of products.
It's just that for large scale power delivery, attempts to have a floating system have failed or have been impractical. For many reasons unimportant here, they can't
avoid grounding, so they need to ground properly. But grounding of the pipework, for example, is exactly what
causes the deaths; contrary to popular misconception, it's not a safety feature against electrocution, it's
the opposite of a safety feature. From the fact that the pipework needs to be grounded, comes the absolute requirement to ground
metal cases of the devices to the same potential, so that if a live wire gets loose inside that equipment, instead of applying a voltage to the case, it blows a fuse, so that you can't accidentally touch the pipework and the case of the broken device at the same time, and receive a shock. If
nothing was grounded, there wouldn't be an issue, either - this can be locally achieved with isolation transformers.
The nice idea was that by mandating (by law), that any device would have its metal parts connected to the same earth pin, such incidents would be prevented. But products do fail, and people fail to follow the regulations. So in some cases, the "protective earth" is a very actual deathtrap. The name is misleading.
It wasn't until about late 1990's this problem was
finally fixed: by introducing RCDs. They completely changed the game, now the "protective earth" suddenly isn't a deathtrap anymore. Still some 30-40 years ago, dying of electrocution was, although rare, a real cause of death. Now it's approximately comparable to being caught by UFOs. RCDs play a big part in this change. (In this country, for example, the death rates went down from 20/year to 0/year within just two decades. During the same time, traffic accident fatalities went from around 800 to around 400.)
So by giving the advice of carefully grounding every metal piece,
you are first creating the safety issue, which you are then solving by adding an active protection device. After all, with a car battery and an inverter, a truly floating system is a very actual possibility! So now we are back looking at the same issue they had to look when designing electric distribution and grounding. Can you have a floating system? Or do you
need to ground, and then add protection against the risk it created? The answer should be in the user manual of the inverter. For example, if the inverter is not isolating at all, then you could receive a shock by being between a loose live wire inside a faulty device, and your car metal body. In which case, you need to tie the case of this faulty device to the metal body of your car, so that a short circuit current flows and blows your cheap inverter, instead of allowing electrocution. Or, even better, highly recommend, add the RCD. The answer should be in the manual, anyway.
By all means, use an RCD, it's one of the best things to happen since sliced bread. In a floating system, there just isn't any mechanism for it to act, because the type of fault it's protecting against cannot happen. But trusting a forum post deduction isn't always a wise thing to do, and if we miss something, the RCD can be valuable. It's not going to hurt (as long as you are able to prevent the "false sense of security" phenomenon).
All in all, if you buy certified, somewhat known brand products, and use them as intended, the risk of electric shock is almost zero today. This includes connecting an inverter to a battery and powering things with it. Regulation exist for a reason.
What comes to inverters, it is completely impossible to try to google the matter, because people, inverter manufacturers included, completely mix up the concepts of "grounding", with at least three totally different meanings in this context. The only relevant question here is, is there a path for current to flow from either of the L/N wires, to the battery input terminals. If there is, then for a safely floating system, the battery, and if it's in a car, the whole car itself, must be protected from touch by isolating material. But if the inverter isolates, it's trivial to build a floating system.
Forbidding the word "ground" by law would probably increase the safety more than any other mean.