'Play your way' is the gaming industry's favourite phrase to spout out whenever they want to appear like their game has choices. But rarely is that the case. Most of the time being able to play your way amounts to having a choice as to which gun you can use. Or whether you use stealth or go in guns blazing. Maybe, if you're very lucky, you can choose which door to use.
New Vegas, however, laughs in the face of those choices and lets you do what you want. If you want to side with a faction, you can side with that faction. If you get bored, you can betray them. If you feel like it, why don't you stroll into the enemy camp strap their leader up with explosives and entertain the wasteland with a fireworks show?
It's probably best explained by the fact that you can completely ignore all the factions and conquer New Vegas yourself. If you can think it, the game likely allows for it in some way.