I think a game that sticks out as my very first character oriented game have to be the original Suikoden for Playstation. It was one of those experiences where you play it from start to finish in a marathon fashion for a week straight. I had summer break from school and all the time in the world. Within that game there are 108 stars of destiny but truth be told there's only a handful you'll eventually bond with one way or the other. The core group of them was always interesting, such as Gremio, Cleo, Victor, Flik, etc. etc. but I always had a soft spot for Odessa, the leader of the Liberation Army. The scene when she dies to this day still brings tears to my eyes, though possibly a lot having to do with the fantastic music. I was really bummed out that she died and it also sort of marks the start of the real adventure.
Following her was probably Snake in Metal Gear Solid. Sure, Meryl, but the fact that they gave Snake a personality, a set of morals and principles and ideas really made him likable. And the fact that I could relate to him as a loner made it all the more fitting. I to this day still enjoy the exchange between Meryl and Snake because he reminded me so much of myself at the time;
Meryl: Is there anyone you like?
Snake: I've never been interested in anyone else's life...
Meryl: So you are all alone. Just like Mantis said...
Snake: Other people just complicate my life. I don't like to get involved.
Meryl: ...You're a sad, lonely man.
The girlfriend at the end of Shatterhand. You fight the whole game to rescue her and finally face off against a giant robot which you, as you have done with every other enemy you come across, tear it apart with your augmented robot hands. Then, through the rubble, you see the body of your girlfriend as you arch nemesis explains that she was inside the robot he whole time and you just beat her to death.
Final Fantasy IV's Polom and Porom. They turn themselves to stone in order to save your group from a trap. My brother and I went back to that screen dozens of times trying to figure out a way to free them.
I remember reading a Nintendo Power article about a guy tearing up over the ending with Shadow on Sonic Adventure 2. I think that, in turn, gave me some kind of sympathy over the hedgehog. So there, I guess. I remember feeling kind of bad for Tim in Braid when it turned out he was this super-obsessive shitty guy, otherwise.
Thinking back it's only recently I got really involved in caring for characters, I LOVE the metal gear franchise but I wouldn't really care if any of the characters died.
For me it's probably The Uncharted trio of Drake, Sully, and Elena. They were some of the first "human" characters that are just so well written it's hard not to care for them.
Atlus does a great job of making me care for characters. Chie, Yosuke, and Naoto from Persona are awesome and Vincent and Erika from Catherine are great too.
And as you say Clementine is one of the only characters in Walking Dead I really care about. I like Carly too but that might be because shes the only other loyal member in the group. I did care about Kenny before he became a huge cunt at the end of episode 2.
I'm probably missing a ton of characters but these are the ones that come to mind.
The cast of Chrono Trigger (cigarette-burning-scenario answer: Robo. When his brothers fuck him up, WHEW! That was hard to watch...).
Celes in FFVI probably. She is a tragic lady, and it probably helped that I identified with Locke the most.
This is a very close second.
@PhatSeeJay: I don't get the love for Clem. I just find her annoying or only there for when they need a cheap "WILL SOME ONE THINK OF THE KIDS!" moment in some scene.
Can't say I would call it cheap, if anything I would call it the first proper source of conscience for me. Because it's the first time I have regretted murdering someone in a video game for the simple reason that she sees it.
The other kid(s) I don't give a flying toss about. It's just her. She wants to help, don't get into any unnecessary trouble that most other kids does and she has saved Lee's life. Gives her a lot of credit in my book.
It's hard to remember that far back. I remember feeling for Link in OoT, I wanted to beat the game because Link needed saving and so did the rest of Hyrule including each area and it's inhabitants. Link was just a kid in a fucked up world and instantly he becomes an adult without the time in between so he didn't have any of the experience of getting older. I especially felt bad for the people in Kakariko Village because their town was so small and shitty but their graveyard was so big. The second you see the zombies near Hyrule Castle I knew shit was bleak.
Really, besides characters that I spent a lot of time building up in RPGs because of time investment, the only one that comes to mind is Clementine.
That pitchfork moment in episode two has pretty much made The Walking Dead the game to beat for my favorite of the year. It actually intentionally made me feel regret, which isn't an emotion I recall ever feeling in a game before.
Hrm. I'm trying to think.
I played Super Mario RPG when I was 12 or 13, and at the end your friend Geno returns to the stars, leaving behind the doll he used. That shook me up as a kid, the idea that your friends can just leave and you'll never see them again.
I don't think I actually got that attached to the characters in FF6 when I first played it, not in a personal way. I thought they were cool but I couldn't really empathize with them yet.
It was when I was 14 playing Final Fantasy 8. It wasn't even Squall and Rinoa (though an introverted teenager spending all his time being miserable in his room certainly was a reflection), it was Selphie and Irvine. I wanted them to be happy with each other, I think that's the first time that's happened. Being pleased that the characters are happy.
I think that's a big thing BioWare figured out. It's not always so much that people want their Shepard to get with some characters, but they want to play matchmaker and talk EDI into dating Joker, or find out Garrus and Tali hooked up.