People are sad that so many games are all straight white guys! So here are a few that aren't.
Upfront Warning: this is all my tastes, so no first-person shooters or, really, anything where you walk around realtime, since they make me nauseous. Also nothing not in English or not playable on Mac. It will skew heavily towards various types of interactive fiction because I like interactive fiction.
Fallen London creates a truly astonishingly vast steampunk/cosmic horror/comedic/I-don't-even-know-what world as a free-to-play text browser game. It's amazing. Hang on, let me find something non-spoilery. Ah! Here's a snatch of journal entry from the Iron Republic:
' - insisted that the Republic was more a human institution than an infernal one, inspired as it was by the Surface revolutions of fifty and a hundred years ago. For if a law of men could be torn down by a mob, why not a law of nature?'
Side characters come in a variety of gender-occupation combinations, you can romance people of whatever gender/s you choose (generally a male and female romance option are presented, and you can pick one or both or neither) and one of the choices for your own character's gender is, famously:
My dear sir, there are individuals roaming the streets of Fallen London at this very moment with the faces of squid! Squid! Do you ask them their gender? And yet you waste our time asking me trifling and impertinent questions about mine? It is my own business, sir, and I bid you good day.
A note: do not pursue the Name. It is legitimately a Bad Thing. I know that games warn about things, and it's just a thing that people ignore, but. Oh. My. Gosh. Really not good. Do not follow me should I go North.
Or anything by Emily Short (yes, a pseudonym), because she is so cool and I have a total authorcrush on her. But Bee is most accessible.
Bee is the first realist thing on this list. It's an unusual story and protagonist for a game—the story of a homeschooled girl studying for spelling bees, and the life that happens through or around that. Even if her life isn't exactly one that you identify with, it's one you can feel. The father in particular at least once I really really wanted to punch so hard, and at least once just wanted to hug and not ever ever ever let go.
If you play this story, you may never see those pages that made me feel those things, but you may see others that I haven't.
Bee is essentially a choose-your-adventure type story, but much more fluid; it takes advantage of the medium to circle back around, present or hide options depending on how far you are, and so forth.
Don't be fooled by the title! This is really more like a fantasy court-intrigue stats-heavy choose-your-own-adventure. Choice of Romance builds a nice little world where you can shoot fire and play politics and romance the Queen/King. I haven't played the other two games in the trilogy, but I certainly liked the first.
The Choice of Games people care about gender stuff. They are cool like that. Also they have a game where you can be a dragon.
You may have heard about this one when the IGF Grand Prize winner was all "oh hey I got this booth but imma just paint over this and make it a howling dogs booth."
Porpentine, like Emily Short, is someone whose stuff feels like they really know how to use their program's abilities to the game's benefit. In this case, Twine. Which, if you don't know what that is, means hyperlinks, basically.
This game is definitely the most surreal thing on the list, but you're definitely imprisoned. At one point I think you're Joan of Arc, or at least dreaming that you are. It touches on dreams, being trapped, being female, being important, being unimportant, just all sorts of things, really.
found some good planets today
I have a totally irrational love for this tiny little Twine game. It starts like one thing, and then you're all ooooooh, I see. Ends as something some people will identify with quite strongly, I think.
Just realized these were all interactive fictions to some extent! That won't do.
The Submachine games are Flash point-and-click puzzle/adventure games in the style of Myst and such. Mysterious sinister and beautiful places and so forth. There's a storyline that progresses as the games go, though they're not done yet!
Counts as good-enough-for-the-list by not doing anything to the contrary and having a gender-unspecified protagonist.
Included because its appropriate for Halloween!
Even scarier than the previous.
ETA: Please add other games you know of! I just sort of wrote games until I got bored of writing games; this is a criminally short list, really.