He is opposite her (because he's on the other side of the table) He is the opposite of her (because he will eat no fat, whereas she will eat no lean)
Centuries ago, both these senses were actually more likely to be expressed using the preposition to. This NGram shows how to has simply been discarded for OP's sense of facing, on the other side, and this one shows how it's been replaced by of for the more figurative complete contrast sense.
"He hid it somewhere between the back door, the shed, and the oak tree."
"Among the meals that we had, several stand out as exceptional."
"Police paced among the crowd."
The above examples are from the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language, page 636. Also, there's these excerpts from that page:
There is a well-known prescriptive rule saying that between is required when the complement denotes a set of two, and among when it denotes a larger set. This rule is based on the etymology of between, and is empirically quite unjustified, as is now recognized by most usage manuals.
The difference between the prepositions is thus not a matter of the size of the set denoted by their complement. It is, rather, that with between the members of the set are considered individually, whereas with among they are considered collectively.
Such verbs as choose, divide, share can accept both prepositions, though the collective interpretation of among will normally require a set of more than two: . . .
A decent usage dictionary, such as Merriam-Webster's Dictionary of English Usage, will usually have an entry on this issue (e.g. "between").