The definition of a branch of photography as "Fine art photography" has nothing to do with the useless "but is it art?" question as infinitely proposed on internet fora.
Photography is practiced by humanity mainly with some practical purpose. Imagine a product catalogue, a leaflet, a brochure, the photograph on your identity card, photographs to classify insects, to draw an inventory of a museum, pictures to illustrate gossip newspapers, to illustrate yesterday's foot-ball match, to document a war, or a famine, or a speech from some politician etc. etc.
When we say "photography" we include scientific photography, documentary photography, news photography, paparazzi-style photography, war photography, fashion photography, legal and forensic photography, pornographic photography etc.
A very small percentage of all photography produced has, instead, the purpose of creating something nice to hang on a wall. There must be a term to distinguish this "candidate to wall hanging" photography from all other kinds of photography. This term is "fine art photography".
Think of it as "a photograph created with the same purpose of a painting".
When people say "Fine art photography" they just mean "a photograph intended to be nice and hung on a wall". They just mean the purpose is only being pleasant to the eye without any practical further use.
By the same token a fine art ceramic plate is a piece of ceramics that you hang on the wall (because it has a nice colourful pattern) and is not intended to be used to eat soup. Since the times of the Etruscans and Greeks there are vases, kraters etc. that had no other purpose than adorning the house or the garden. They never saw water or wine in their life. They were "fine art kraters". Maybe they only had a geometric pattern on them. Their reason to exist was just aesthetic and not practical.
That said, IMHO "fine art photography" is not "art" in the highest sense as no photography is ever "art" in the highest sense, not even Saint Ansel's production. I see it more as a craft, the domain of skill and taste. Mestiere.
But it is legitimately called "fine art photography" because that is the linguistic convention to distinguish it from the other genres of photography.