Is it necessary to do all the story for every character?
It depends on what you want to do with that character, if you are not completing the story, you wont get acces to some of the milestones which get you better gear at the end.
And without the milestones you are going to get stuck at around 265 to 270.
Also if you complete the story you are getting access to the exotic questlines.
If you want that character just for pvp, it is not really necessary to complete the story, if you want to have some progress, you should complete it.