Gonna concur on Saints Row IV. I came in once to find my friend playing it (either that or Saints Row the Third, which IV beats out handily) and within a couple of minutes surmised without a word from him that he had had a really rough day/week and the game was helping him unwind. Before that realization, I had never considered that these "play the guy who steals cars and shoots random people with impunity" games could actually serve a (potentially) beneficial purpose.
I haven't actually played much of it (though I now own it thanks to a Steam Sale), but the game line is very sandboxy, and although there are missions to go on and characters to interact with, you are pretty much free to run around gunning down baddies, or steal a car and drive rampant through the streets. Or (eventually) a helicopter, or a tank. Or... not sure if this is in all the SR games, but I had a TON of fun with the "insurance scam" quest, where you throw yourself into traffic and try to get ragdoll-physicked for as much moolah as possible (the farther your body flies, the better). I did not expect to enjoy that quest but man, was I wrong about that.
Saints Row IV's improvement over the previous ones is in the addition of a sort of super powers. When I saw my friend playing it for the first time, I thought it was something you probably had to earn by playing through a lot of the game, but he said you get them almost immediately.
Another recommendation, not so much about shooting though: Go on Kongregate and look up Earn to Die. I was just idly browsing games one time when I felt really depressed, and came across this one and thought "Looks stupid, but why not?" Five minutes later I was mowing down zombies with my monster truck, doing no more than pressing forward and occasionally up and down for stability, and laughing my ass off. There's something to be said for dumb fun.