Why is the movie titled "Independence Day"?

The 4th of July holiday in America is called Independence Day, celebrating the day the Thirteen Colonies made themselves a new nation, free from the British.

So from a historical point of view, it's quite logical whilst the national holiday is called Independence Day.

However, in the film they twist this a little. Consider the president's rousing speech towards the end of the film:

Perhaps it's fate that today, July the Fourth, we will once again fight for our freedom. Not from tyranny, persecution or oppression. But from annihilation. We're fighting for our right to live, to exist. From this day on, the fourth day of July will no longer be remembered as an American holiday but as the day that all of mankind declared we will not go quietly into the night. We will not vanish without a fight. We will live on. We will survive.

So the President here is referring to Independence Day as primarily symbolising the day the fight for America's freedom was won. He then relates it to their current situation by showing how they're no longer fighting for freedom, but to avoid annihilation. But his point is, it's still a fight.

However this time, it's not American's fight against the British, but the world's fight against invaders - so when they win, the world can celebrate the day together, because the world will have been saved.

Of course, it doesn't quite make sense for it to be called Independence Day, as they're not independent of anything as you correctly surmise. However, given how the President relates it to the existing Independence Day coupled with the fact these events are occurring on 4th July, it makes sense that he is inviting the world to celebrate winning the fight on that day.

In the movie,aliens have destroyed most cities of world and mankind's best weapon is the will to survive. I think the survival of humans in this movie is called 'Independence'

Here is what Wiki has to say about your question justification for commemorating the 4th of July as a worldwide "Independence Day":

With the successful implantation of the virus, Whitmore leads the attack against an alien destroyer approaching Area 51. Although the force field is deactivated and the fighters are able to inflict damage, the hull of the destroyer is too big to inflict serious damage. As a result, the fighter's supply of missiles quickly becomes exhausted. As the destroyer prepares to fire on the base, Casse has one missile left, but it jams. He decides to fly his plane directly into the alien weapon in a kamikaze attack, which kills him but destroys it. The Americans inform resistance forces around the world about how to destroy the alien ships. The nuclear device destroys the alien mothership as Hiller and Levinson escape unharmed back to Earth. The whole world then celebrates its heroes' victory as well as its true 'Independence Day'.

The movie is a summary of the events that transpires on the day of American Independence; when an army of Alien beings attack the planet looking for a new habitat. The defenders of planet earth fight valiantly to save humanity from extinction and win back their independence from their evil counterparts of the galaxy. Hence, the title aptly describes the plot of the movie.