This does not look like vector art to me (possible exception - the three tabs on the lower left of the screenshot with the boats). What it actually looks like to me is 3D art with somewhat pixely textures and some 2D elements overlaid. 3D art is similar to vector in the way it can be exported at different resolutions. You wouldn't usually see a "dirty-looking" texture like the cobblestones in the first image made in vector art just because that style of coloring would take a silly amount of time compared to something like cell shading for which vectors are more naturally suited. The vehicles, stars, and explosions however would be easy to do in vectors. But the water effects in the boat screencap and the shadows on the vehicles say 3D to me.

As sunandshadow said, that probably isn't vector art.

The textures are probably hand-painted, though it's not really pixel art in the traditional sense. ie, they probably used painting techniques rather than pixel techniques. It's a fine distinction, true.

For hand painted stuff, you might do some thread searching over at polycount. For example, here is a thread on hand-painting rock: http://www.polycount.com/forum/showthread.php?t=89131

With work, you can hand-paint a texture then do some creative editing to make it tile seamlessly, as the ground in your posted screenshots tiles.

For the other stuff, the objects and the like it is possible that they are 3D rendered, but they also look like they were maybe also hand painted. You could do it either way.

Now, if you need to provide sprite graphics for differing resolutions there are a couple ways you can do it:

1) Provide different sizes of sprites.

2) Scaling

If you provide different sizes, you increase your workload but the sprites can look better since they are designed for the given resolution. Scaling can cause things to become a little bit blurry (scaling up) or smudge out details (scaling down). However, looking at your posted screenshots, I suspect that they are actually scaling. The sprites are not extremely high detailed, and some portions of them (especially the planes) seem to be a bit blurry.

With sufficient practice, you could duplicate this look in Photoshop or, if you don't want to pay for Photoshop, the Gimp. Hand-painting takes a bit of practice, though, so don't expect to recreate that look right out of the gate. You could also recreate it using low-resolution textures painted onto 3D models, and then rendering the models, again at a relatively low resolution, perhaps with some color processing to more closely emulate the pixel-art look. For this, you could use a combination of Photoshop/Gimp plus a 3D modeler such as 3DS, Maya, or Blender (the free alternative).

Most 3D programs include functionality for tiling textures, so yes you can absolutely take a pixel art texture or any other kind of texture and tile it across a surface in a 3D program. It's quite common to see tiled textures on terrain in 3D games. Some 3D programs also include functionality for importing vector images and using them either as a texture or as a curve to be extruded into a 3D object.

Anyway, I'm sorry if anything I said was confusing. Though it's true we don't have a big population of artists that hang out here. What I really think beginner game artists need to understand is that you can create art that looks pretty much the same using any digital art method. If you picked out three more example games that you thought looked the same, they would probably turn out to have been made with a variety of methods. Some things take more time or skill to do in one technique than another, but there's someone out there who can do it. I'm saying this partly in reference to what JTippetts points out about painting shadows - I didn't consider that possibility because it's beyond my own skill lever at painting, but I know there are people who do have and use that skill. I'm also saying that partly in reference to the fact that I am mainly a vector artist, and if someone wanted to pay me to make a copy of one of those screenshots using only Inkscape, I could do it, but I also know it would be more time-consuming than doing the same thing with other methods because it's not what vector art is optimized for. So, if I could do it, you could do it too, but I don't want to advise you to use a method that's not the most efficient.