John tells Sherlock that he is John’s best friend and after that Sherlock begins to refer to himself as such.
I never expected to be anybody’s best friend. / Your best friend is a sociopath who solves crimes as an alternative to getting high.
It is important to Sherlock. He is not just any friend: he is the best. But the weird thing is that he never calls John as his best friend, which would be the logical thing to do. Not once does he say My best friend John. It’s because Sherlock sees it differently.
The best is superlative: good, better, the best. In order to use superlative you need to have a group of at least two items among which you choose one to be the greatest. If you have only one there is no need to call anything the best. The value of the one and only comes from the uniqueness of it.
I don’t have friends. I’ve just got one.
John Watson. My friend, John Watson. John.
Sherlock uses the word friend like other people use the words partner or husband: like there can be only one in the world. Sherlock’s world of friendship is monogamous and The Friend of his is John.