No, "остановиться" does not refer to stopping a vehicle specifically.
"Остановиться у родственников" = To stay with relatives (e.g., during your vacation)
"Часы остановились" = The clock has stopped.
What you hear from Russian-speaking mothers is one of the imperative mood verbs used to deliver the "stop it!" idea.
Перестань! Прекрати! = Stop it! Quit it!
Успокойся! Угомонись! = Calm down!
Хватит! Довольно! = Enough!
Хорош! Всё! (colloquial) = Enough! That's it!
I might think that "пререстань!" has a sense of "stop and don't do it anymore" while "останови!" (more often "стой!" or "стоп!") has a sense of "stop now but then continue after whatever"?
The root of the verb "остановиться" is "стан-", which has the meaning of "being stationary".
"Пристань" = a pier (a place for ships to be stationary)
"Стан" = a camp (one of the meanings for this Russian word; a place where people stay)
The "о-" prefix in this case delivers the meaning of "becoming, making".
"Остановиться" = to stop (to become stationary)
"Остановка" = a [bus|train|taxi] stop (a place where bus|train|taxi become stationary)
It's a prefix, not a suffix. Here's a link that you might find helpful. https://en.wiktionary.org/wiki/Category:Russian_prefixes
it has a lot of meanings. In this case, it doesn't have a particular meaning.
Does this sound as passive-aggressively rude in Russian as it does in English?
I wouldn't call it "rude" in either language. However, I agree it would be more polite to add the word "please."
For example, in a taxi it is enough to say just "вы можете остановиться здесь?". It is enough polite, whiteout "please". No polite is "Эээ, слышь, тормозни здесь".
Can't this mean: "can you stay?" It was not accepted. OlegK's first example below suggests it should be accepted.
I also wrote Can you stay? Why is it wrong and what would be the word for "stay''
Stay = "остаться/оставаться"
Can you stay for the night? = Ты можешь остаться на ночь?
Just like in English "Can you stop?" and "Can you stay?" mean quite different things, so do "Вы можете остановиться?" and "Вы можете остаться?" in Russian.
However, if you want to say "stay at a hotel" or "stay at my friend's house", it will be translated into Russian as "остановиться в гостинице" and "остановиться у друга/подруги". That's one of those language quirks, sorry.