How can an app tell what vehicle it is in and what qualifications the driver has?

Waze has settings for this.

Kind of my point, it relies on the user setting something.

Not really a problem though. It's trivial to ask.

Might be true in a haply belief but I give you one first big critique:

It's NOT easy to time the release of such a feature.

In fact, since this is not easy, I see a big opportunity for a competitor to Waze.

do they have "how tall is your rental truck?"

I had a friend I helped move and we had to deal with google continually routing us towards low bridges.

You can buy satnavs with that feature: https://www.tomtom.com/en_gb/sat-nav/truck-sat-nav/ and of course, many truck rental companies will also rent you such a sat nav.

Does it work 100% of the time? Is it mathematically disproven, that it won't fail on any occasion?

Is it mathematically proven, that every bridge is always appearing in the dataset after it is possible to drive underneath it? Or over it?

No, having a sat nav doesn't obviate the need to be attentive to road signs and the size of your vehicle.

No map data supplier warrants that their data is free from errors, and even if they did the data could be out-of-date within hours as roads are being built all the time.

There is no proof - mathematical or otherwise - a sat nav won't switch off if you fail to plug it in. Neither is there any proof a bridge won't be closed due to roadworks, an accident, or other reasons.

How about drones?

In OsmAnd you can set vehicle parameters: height, weight, and width limits.

you could probably just tell it your vehicle and/or qualifications if it was important enough,

True, maybe by scanning the number sign plate.

This is especially important in Europe where we are having more and more city centers reserved for inhabitants, or forbidden to polluting vehicles.

Is it? I think we can decide for ourselves.

I don't know; I rented a truck in an NYC outer borough for a move and had the hardest time navigating through the city because I had no easily accessible way to figure out where trucks were allowed.

I stopped a car in a local forest because the street there is not legally accessible for cars, which don't have a special permit.

They gave reason: navigation system.