It is indeed difficult to find information about this if you can't read Japanese or Korean. Most ferries are passengers only.
I ended up asking a friend in Japan:
There is at least one car ferry between Busan (Korea) and Fukuoka (Japan) run by KoreaFerry. The information I got says that the cheapest room category is included in the car price, but for the driver only.
This means that at least for this ferry, additional car passengers have to pay extra.
Whether that additional fee is the same as the single passenger fare I don't know.
Also this doesn't mean there are no other companies out there with different conditions but I pretty much doubt that.
The question to ask would be 'Are there any ferries in the world that charge per car including all passengers?'
Edit: My second source in Fukuoka came back to me: There is another ferry between Busan and Shimonoseki operated by 'Pukwan'. Again it seems you have to pay by passenger. Also for the ferry from Busan to Fukuoka you have to buy a return ticket for your car. There are no one-way tickets available. This shouldn't concern any hitch-hikers though.