What style of art is this? This art style is very reminiscent of old cartoons. Could you tell me if it has a name? I've become very interested in it.
The style of 20s-30s cartoons is called Inkblot (see http://tvtropes.org/pmwiki/pmwiki.php/Main/InkblotCartoonStyle for more details), maybe this could help in your research.

The two images you posted have both an Inkblot feel, but they are more "graphical", in a sense of being more graphic illustrations. Probably they're also both vector illustrations, which gaves them the feel of being like "plastic". These are more personal observations, of course, but they might help you.