The interesting thing about Star Trek to me is the idea that our real-world political ideologies don't even apply in that universe. Our politics are built on the different ideas of how we distribute resources (what we typically call wealth) - but Star Trek is a society with cornucopia technology. They don't have wealth disparity, because they have fundamentally unlimited wealth. It's a completely post-scarcity economy where our economical models make no sense, and neither do our political structures.
From a writing standpoint I'd call it left-leaning since it often deals with the contextual issues of social inclusion and in/out groups, but a lot of that is entirely subtextual to the stories being told. You can largely ignore the implicit politics of Star Trek and still have a good time watching it, if you're inclined. It's also worth saying that no one is really a political island in and of themselves, no matter how much they try to be. We all have both the fundamental thinking of the political right and left ingrained in our thinking, to some degree or another, and where we may disagree with one on the macro level we will often resonate with it on the micro.