I think that when it comes to focusing on character, the games that actually involve elements of a character in the play processes tend to facilitate this a lot more than games that do not, or that do so poorly.
I have in mind two recent games that I played in. I tend to GM most of the time, but lately the players in my home game have been rotating as GMs and I’ve been playing. Then I also play in an online game with
@Ovinomancer that’s run by
@Manbearcat. The home game is 5e D&D. The online game is, as has been mentioned, Stonetop.
In the 5e game, I played Teemo Nines, a nine-fingered ranger with the folk hero background. He was once a scout in the queen’s army, but was captured and tortured (they severed the middle finger on his right hand). He would have been abandoned to his fate except he was included in a prisoner exchange between the queen’s army and their enemies, in order to ensure that a minor noble officer was returned. The experience soured him on army life and authority in general, so he went awol, and struck out to the frontier. He became a folk hero after standing up to soldiers of the duke. He’s got a pretty strong Robin Hood vibe.
He has two traits that I think are relevant. “I am an arrow” meaning he acts swiftly and directly, and “I hate to see authority abused”, which means he’s not likely to put up with tyrants, bullies, and the like.
A possibly significant point is that I crafted this character with the gist of the game in mind. Our group had discussed what we wanted to see in play, and what kind of setting we wanted. We went with a kind of low-magic, quasi-historical iron-ageish type setting. The starting region was to be a frontier with lots of room for exploration, but also a kind of contested region where one country holds sway, but where it is closer to another country that has designs on the region. So the possibility for some political intrigue type of stuff, too.
So I crafted my character with this setting in mind. I chose his traits and his background precisely to set up potentially interesting situations. This combo of character and setting would seem to have potential for some drama, I’d say.
In play, it was not always meaningful. Sometimes if was. At other times, it didn’t matter at all. When we came into conflict with the duke’s men, it would come up. The fact that Teemo was a deserter mattered in an early situation, then eventually got resolved in a very handwavey way. It seemed that his status as a deserter was a bit inconvenient to keep things moving for the group, and so he was pardoned as soon as events had made that even remotely possible.
After that, it mattered far less. My attempt to use my Folk Hero quality to gain shelter/protection from the commoners worked sometimes, but not others. Depended very much on GM whim. The further into the campaign, the less any of Teemo’s traits and background mattered. Then one GM decided to run Wild Beyond the Witchlight, and any possible character focus was abandoned. I could still portray Teemo in a man er consistent with what had been established, but calling play character focused at this point would be a very weak claim.
I suppose many people would cite that sone of this can be attributed to poor GMing. And while I get that, I don't know if that’s quite it. The GMs (there were three across the length of the campaign, though one had only a very brief stint of two sessions) mostly didn’t do anything that 5e would say “don’t do that”. The game itself doesn’t do anything to steer play toward character focused play. If it happens, it’s mostly independent of the actual game. The Traits and Backgrounds suggest such play, but they can be totally ignored.
Compare this to my Stonetop character. Interestingly, the basic premise of Stonetop is somewhat similar to that of the D&D campaign. The PCs are all inhabitants of the town of Stonetop, and they hold positions of importance there. It’s a town on the edge of civilization, and is a low magic, iron ageish type setting. My character is Cullen, the Judge, who believes he was chosen as a prophet of Aratis, god of law and civilization. In addition to his Background as a Prophet, he has another relevant trait, his Instinct, which is Harmony. He thinks everyone has a prt to play, and in the greater good.
So far, play has absolutely been focused on Cullen’s traits. Every situation that the PCs find themselves in, their instincts are put to the test, or their backgrounds are relevant. The text of the game tells the GM to do this. Play should not just be about random adventures in this frontier space, where one PC could be swapped out for another with little impact to play. The events are about, and reveal, the characters specifically.
And I think this is part of the challenge with some of these discussions. Someone will describe a game like Stonetop, and someone reading that description will have experienced a game of 5e D&D (or similar) that seems similar. The 5e games that I run tend to lean that way. But there’s a significant difference. One is designed to deliver a somewhat broad possibility of play experience. The other is designed to deliver a specific and deep experience.
To sum up, 5e said “sure, you can do that, but you’re mostly on your own” while Stonetop said “not only should you do that, you must.”