We would need a higher dimensional political process to apply it to, first. A two party system can only describe one-dimensional politics.
And as long as we stick to this incorrect single dimensional model, we make it even more difficult to talk about 3rd parties, exactly because some (many?|most??) of them don't fit the simplistic left/right continuum. To continue with an earlier example, American libertarians do not fit cleanly on either side of that model. I'm pretty sure there are other examples, but the Libertarian one is the one I'm most familiar with.
It's something of a chicken-or-the-egg problem, granted, but right now it's probably easier to start changing the language we use, than it is to overcome first-past-the-post single-member-district voting systems, etc. [2]
It's not a two party system, but it is a two-party dominant system.
That is a puzzle that should be teased apart.
Realistically, the terms "left" and "right" don't mean a damn thing in modern politics, at least not in the US. There's not really a cohesive ideology that you can say is "the left ideology", or one you can call "the right ideology". Both labels are, at times, attached to a smorgasbord of ideas, some related, some unrelated, and probably some that are diametrically opposed to each other.
Mostly those words exist simply as pejoratives now.
And to the extent that they have any meaning, the idea of one dimensional model of political orientation is brain-dead anyway. You can't actually reflect the variety of political belief like that. Take, for example, Libertarians, who are, as the old joke goes "left of left, AND right of right". That shouldn't be possible, and yet, here we are.
We really ought to quit talking in terms of "left" and "right" and find a richer and more consistent model to describe political orientation.