For example, can you say 'I have more than one best friend' and is that grammatically correct given that these friends are equally good?
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The term best is a category. How many units fit in the category depends on context.
If there is a contest, like a dog show, there may be a best in each breed, and a best in show. Overall, the show would issue several best medals.
Stores often categorize products based on features. They may label several of a given type of item good, several better and several best. Whether they have one or more in each category mostly depends on the depth of their offerings.
The term best when applied colloquially is a high end compliment, not intended to formally rank a person with regard to all friends or aquaintences. It is likely that your spouse would not be annoyed or feel diminished if you thank a friend for an assist by saying
As to best friend, it again depends on context. You might say
There may be several individuals you could say that about without confusion.
However, when you say
most US speakers would probably think that the identified individual was likely the single person you felt closest to (aside from relatives, who might or might not be labeled best friends).