Books, theater plays, dramas in general. Humans have been able to feel sorrow, anger, delight over fictional plays for thousands of years. All it takes is some weird combination of writing, narrative and possibly acting that we can connect with, or find to our liking. It would be weird if video games made slice of fiction somehow unable to achieve this.
I've cared about main/NPC characters of the Last of Us, Walking Dead and KOTOR.
Of course, we care greatly of our *own* characters with ease, and that doesn't necessarily require one slice of good fiction. They are of us and turn parts of us, or at least into our property in more meaningful way than NPC companions or suches. In single player games, even in RPGs of epic scale, this doesn't necessarily become a big deal. We play our ME or Skyrim for 50 or 100 hours and call it a day. Character gets retired, curtain gets called. If we return to the came, we prolly start a new character.
For multiplayer games, specially for MMOs, it is different; There are millions of people with over 10 year old WoW characters that are never retired. I'm sure most all take few year breaks from the game here and there. Yet, if/when they return, they often return to the character,too. Characters who have kinda lived inside the person for a decade. BLOODREAVER the orc warrior got rolled when uhh..Jim was 14. Thus the cheesy name. The orc lived inside Jim as he graduated from High School and College. BLOODREAVER has been faithfully dinged as Jim's first born was born, as marriage between Jim and Jil got made. It can be pretty huge, when you think of it. This is much, much bigger than some NPC companion saying sweet nothings about how awesome Shepard is.
..I realized there are Ultima Online players who prolly have older characters than many of the people reading this message are. Heh.