I tend to agree that romance is typically handled pretty poorly in most mediums, and can be especially awful in films. Because there's such a wide variety of games, it can be hard to generalize.
While romance can be bad in video games, I feel that overall, they're pretty well done, at least in the games I've played (yes I know I said they can be hard to generalize, shut up!). Though there are some good examples of romance done wrong, such as the aforementioned the marriage system in Skyrim ("Hey baby, did you notice my necklace? Wanna screw?"), there are better examples of romance done right.
I thought that the Fire Emblem system of bonds and romance was pretty interesting. Granted it was extremely obtuse in the GBA games, and I didn't quite grasp the concept in the console iterations as well, but it was expanded upon greatly in Awakening (mostly due to the time travel/children thing), which I thought added quite a bit to the game. The bonds were added to further due to the finality of death, which added a bit of drama to fights that weren't going to well between romanced characters (with a bit of imagination, anyways).
Others have mentioned Bioware's implementation of romance. I thought that it was handled pretty poorly in Jade Empire, but I loved me some Mass Effect romance (those are the only Bioware games I've played, I'm sorry!). Tali is my Mass Effect waifu, which will be pretty awkward to explain to my girlfriend once we finally get started on our run through together (actually, she's pretty similar to Tali personality-wise, and I'm pretty similar to Garrus personality-wise. It's even more eerie when you remember that the two get together if both alive and unromanced by Shepard by game 3... connection??? But, I digress). I can't really explain what it is, maybe I just get into character too well.
People have also said Persona is a good example, but I haven't played those... yet.
One last game that comes to mind is Half Life 2. I know it was never explicitly stated, but I thought that Half Life 2 is unique in creating a meta-romance, or at least a meta-bond, between "Gordon" (the player) and Alex Vance. That's pretty impressive, considering nothing is ever said between them, which is more than most games can claim. Or maybe I'm just a hopeless romantic and there really was nothing there, who knows.
Overall, I think that romance, when role-playing/ getting into character especially, is a very useful tool for adding to the tension of a situation or the intensity of a scene. Is Jaffar going to bite it and leave Nino all alone?? (Heh, the concept of Jaffar dying in that game is hilarious!) Will Shepard finally get it in?? Is Alex going to get ear-raped by one of those floating pill bugs?? WILL SHE NOTICE THE NECKLACE????!?!! These are all answers one asks themselves when playing these games (or at least I do), and I think it adds a lot to the experience.