Who or what organization chooses the genre of a song if it is not apparent.

No one decides an official genre of a song – a song is just widely known as being under that genre.

When you say “who puts it on the country, or blues, or jazz lists?” – what lists exactly are you talking about? In any case, the person or group that makes the list will be the ones who decide what can go on it – if it is, say, a list of jazz songs, then the people who make the list would surely be the same people who will decide what is and what isn’t a jazz song.

It is usually very apparent what genre a song is anyway, especially if you’re just using broad terms. While it is often hard to describe in words what a, say, a pop, alternative rock, jazz, or country song is, you can tell by hearing it – each genre has a distinctive sound.

Many bands will also identify themselves with a particular genre, particulary as many genres have a subculture behind them (take, for example, the difference between the lives of country singers and “emos”). However just because they do define themselves doesn’t mean that all of their songs have to be under that genre, and while it could be considered “official”, you are still entitled to call it something else if you wanted.

No one really “decides” the genre. However the record company generally decides what type of radio station the song gets sent to. Usually they will create a couple different mixes and send a song to several different formats of stations. Back in the 90s when country was crossing over into pop, there were alot of pop mixes of country songs.
The Eagles for example would have a Soft Rock, a Classic Rock, a Country and maybe even a Pop mix of their song and the record company would send a copy to all of those stations in hopes of getting it played.