BEST (singular noun; usually "the best") the most excellent thing or person

I see this as a question of implied sets. best refers to a set of friends.

friends is a plural noun, a set of friends.

Usual and likely is that the friends in the set are all friends to each other. This allows us to interpret the sentence as We are the best {=best set of all possible sets} of friends.

Less usual and less likely is that they might all be friends of someone external as in “we are all enemies to each other but are all friends of the king”; this is both an unusual and unlikely circumstance. Nevertheless, the same analysis as in my previous paragraph may apply.

At face value, this form is not a set form: you find "the best of parents" (ref.), "the best of pets" (ref.), "the best of lands" (ref.), etc., and let's not forget, also "the best of friends".

• “A good book is the best of friends, the same today and for ever.” (ref.)
  (In this sentence "best" stands for "best friend" among all friends, not a person but tantamount to one, the book is more or less personalized.)

You will notice that in all those constructions there is no question of reciprocity, in particular in the last case. Moreover, it is clear that either a person or several persons are involved or a thing (land) and the definition of the noun "best" can be applied as signifying the most excellent parents, pets, land, friends, etc. We can conclude that all these constructions are of the type that answers to the principle of compositionality (ref.), ie, the meaning of the construct is deducible from the meaning of its parts.
It is clear that in the use of the phrase that you consider the deducible meaning discussed above is not at all relevant:
1/ the referent of the complement of "of" above is the set of all parents, pets, etc. (Amongst all the parents, pets, etc. those named are/were/will be/etc. the best.),
2/ according to the meaning in question this is not true and the referent of "friends", is instead the same as that of the subject,
3/ there is an idea of reciprocity.
This makes already clear that compositionality does not operate here and it follows that the phrase must be an idiom. This is somewhat to be presumed from the addition made in the OALD (= very close friends).
It follows that there is no use seeking to give the different parts of this phrase a precise meaning; at least this seems to be now a strong conviction of mine. Not irrelevant to that contention might be the fact that in the SOED the forms "best friend(s)" and "the best of" are found in the section of the special collocations, phrases and combinations.