How is "I'm sick of something" grammatically correct?
The "of" operator seems to denote ownership. The capital of america is DC.
So how does "sick of" or something similar work?
Certainly, when of is used in a phrase connecting two nouns, then its function is one that could very loosely be called ownership: the capital of America, the end of the day, a bunch of roses.
But of is also one of the prepositions that follow adjectives. There are some patterns that may be useful to learn. For example, we generally use of with evaluative adjectives:
It was good of ( clever of / kind of / mean of / polite of ) you to ... .
But most adjective + of combinations are idiomatic. In other words, the combination has to be learned as a single chunk since there is no predictability in which preposition is needed. For example, these two phrases have roughly similar meanings but different prepositions: fond of - keen on.
Some other idiomatic adjective + of combinations are: proud of, sure of, afraid of, aware of, capable of.
6 indicating the relationship between a verb and an indirect object.
expressing a cause: he died of cancer
The cause of your being 'sick' is the 'something'.
Incidentally, being 'sick of' is an idiomatic/ informal expression explained in the same dictionary.
3 [predic.] (sick of) intensely annoyed with or bored by (someone or something) as a result of having had too much of them: I’m absolutely sick of your moods