Probably because cars are personal and the rest are mass transit. It’s not yours, but you pay to get on board when needed.

I have to explain to non-English speakers why you get in a car or a taxi but get on a bus or train.

Because then it would be carboard

Not true. When you're in a car you're "onboard".

Sure you do. It's just not something people commonly say.

But boarding and disembarking a car is totally valid language.

You also ride on a bus or on a ship but you ride in a car

You board a taxi, though.

I'll agree that it comes down to the personal vs. group nature of the vehicle. Maybe also who is controlling the vehicle (professional vs. amateur)

You don't board a motorcycle or a horse. You don't board a unicycle or bicycle. You don't board a kayak or canoe or a jetski or a dingy.

You DO board a stagecoach, pulled by horses. You do board a trawler or a fishing vessel.

I don't board my own van. I do board the intercity bush taxi (the same van with staff and better paint).

Likely all related to use of "board" for ships and devolved from there.

I think it has to do with the fact that in the first three, you can (and do right after entering) stand up (like on a ship, probably where this was first used), while in a car you immediately sit down?

But ultimately this is a trait of the English language, in my first language (German) we "steigen ein" in all of these.

Perhaps it has something to do with trains, buses, and ships being operated/driven by others? You also board an aircraft, a ferry (okay, that's a ship, but still), hot-air balloons...

Most of the time anyone who travels by care are or can be the driver of the conveyance. That is very rare for people who travel by train, bus, ship, plane.

Not with that attitude

You board something you can move around in or on. You get in or on smaller methods of transport.

but you can board (up) a house

(like boarding up the windows)

I'm guessing because it's a process in which you need to show a valid ticket of travel (or buy it) while 'boarding'. Like onboarding or something.

You can walk on board a bus, boat or train. You sit down into a car.

If you had seen my lemon yellow 1983 Cadillac you wouldn’t say that.

Witness me!!

You also can't "squeeze into" a bus, ship or train, but you can into a car. It's a matter of scale.

You used to need to cross a board or a plank to get on the ship or train but you don't need that for a car.