That would be because "be-" is a prefix that is almost always added to verbs and makes nouns and verbs. It means something along the lines of "cause".
It is, by no means, a recent one, so you're bound to come across examples where the verb is no longer distinguishable and doesn't exist in modern English, as is the case with "believe". The most common examples of verbs with the prefix are transitive verbs ("become", "betray", "behead1" etc.), unsurprisingly, considering the meaning the prefix conveys.
It would then follow that "belong" be, in fact, "be-" + "long". "long" is an intransitive verb originating from an older "longen" which meant "to go along with", and "causing to go along with". If "belong" were to be used in a similar fashion to verbs like "believe", the sentence would have looked like
He belongs the car keys.
Instead, the usage is such that the thing that is owned, and is a property, comes first.
Why it wasn't seems to be just a matter of how language evolved, and not something strictly logical, as @ColinFine's answer points out. However, there is no possible scenario the sentence "* The car keys belong him." would be correct, as it would be nonsensical.
1: If interpreted this way, this verb was probably meant to convey "make someone a head", which is an interesting thought!