Actually, Minecraft will run at whatever resolution you size your window to (348x866):
So it's likely that whomever took the screenshot you grabbed, just happened to have their window sized that way.
The default screen size when I start the game appears to be 856x482, which is pretty close to 16:9. It doesn't need to use a standard aspect ratio or resolution. The display size doesn't have anything to do with the texture packs. It's not required to be divisible by 16. A game might have these restrictions if it was 2D and you wanted to fit a whole number of tiles on the screen at one time. However, with a 3D game the display size (in windowed mode) can be whatever the developer decides they like.
When the game is run full screen, it's best to match the game resolution and aspect ration with the resolution and aspect ratio of the screen.