Recognising genre and sub-genre

Is it possible to recognise a genre/sub-genre by listening a song?

A bit of a trick question in my opinion. If you don't know the genre, would you be able to identify it as a genre that you do not know? Perhaps. Sometimes you will pigeon-hole the song into a genre that you are already familiar with. And there is nothing wrong with that. I think that is a major part of music criticism, being able to identify songs with what you are familiar with.

Is there any authority to approve a music genre? On what basis are they defined?

The authority is the music critics, otherwise known as the audience. All though there is general agreement that some genres are performed in specific places or that the audience is expected to react in a specific way such as dancing, sing along or quiet, studious listening. And musicologists, who write academic papers and publish in academic journals, study this interaction and describe forms. What is a waltz? It is lively music with a 3/4 rhythm with the accent on the downbeat, whereby the audience responds by dancing in pairs via a spinning box step on the dancefloor.

Musicologists do step in to provide guidance on basics of genres, but the audience at large steps in to describe it their own way. However, the biggest force that will have specific songs be identified with a specific genre is the publisher and marketers of the songs. You will notice that for the most part when you step into a music store, that they will genres all ready picked out for you and that you can have pretty strong objections as to which recording goes into which category. The ones selling the music will always have a biggest say on what specific songs get grouped with.

(Writing as a metal head)

Is it possible to recognise a genre/sub-genre by listening a song?

Sometimes. There are a lot of genres and sub genres e.g. in Heavy Metal (Wiki) and a lot of them can be recognised, e.g. Arch Enemy, Melodic Death Metal, can be clearly distinguished from Helloween, Old School Speed Metal, from Man'o'War, True Metal, by melodicness, speed, different riffs, voices from screeming to shouting, growling, singing.

With subgenres it's a bit more complicated, because they belong to the same genre and hence share melodic features and may only differ in the lyrics used.

Is there any authority to approve a music genre? On what basis are they defined?

Musical genres mostly evolve and don't appear over night. A good example is Motörhead, a prototype Heavy Metal Band but at the time, they started out, the genre Heavy Metal didn't exist, it still was Rock 'n Roll. (That's where the Motörhead opener comes from: We're Motörhead and we play Rock 'n Roll)