Who decides the genre of a movie?

I wouldn't say there's anyone who specifically sets a genre for a movie as some sort of official thing. The studio will likely describe the movie one way, but writers and magazines and websites may use different or even additional descriptors.

One thing you have to remember with websites is their primary motive in anything is to drive traffic to their site. By adding a large swath of genre information to the details of a movie they've got a page or article or entire section dedicated to, any search terms could ultimately lead to that movie being found on their site, driving traffic to it.

Also, not all websites are run by the same people. Someone may feel one movie is a political drama, while another may feel it's a political thriller.

It could also come down to one's own personal interpretation of the content of a movie, such as a film critic.

I'd say the best descriptor for a movie likely comes from the studio that made it, or the people involved with making the movie itself (screen/story writer, director).

In the end, I would't say to get too worked up over different sites and publications having different descriptors for movies, as there can be any number of reasons this happens, none of them official.

Film genres can be categorized according to the setting of the film. Often, the genre of a movie is obvious. Can anyone argue Titanic is an comic movie, or Saw is a romantic movie?

It's universal, Titanic is a romantic movie and Saw is a horror movie. So no one does really decide; because subconsciously, we, people or critics, categorise film genres in 4 ways:

1. The environment where the story and action takes place.
2. The theme which refers to the issues that the film revolves around.
3. The mood which shows the emotional tone of the film.
4. The manner of presentation (e.g.: 35 mm, 16 mm or 8 mm).

An additional way of categorizing film genres is by the target audience: While watching "The fault of our stars" with my friend, we certainly didn't think it was horror movie. We both agreed it was romantic, but I subjectively classified it as a drama movie and she subjectively classified it as a comic movie also. It got glassified in wikipedia as a romantic comedy-drama film, and this is the genre of this movie based on target audience.

The truth is genre is often a vague term with no fixed boundaries and a lot of controversy. Film theorist Robert Stam challenged whether genres do exist, or made up by critics. He asked:

genres [are] really 'out there' in the world or are they really the construction of analysts?.

People will start categorizing movies with their own subjective terms. For example:

Linda Williams, a film critic, argues that melodrama, horror, and adult movies fall into the category of "body genres".

The editors of filmsite.org argue that animation, children's films, and so on are non-genre-based film categories.

And so on.