Parodius could make it work, the game is wacky, and the racing bit can be pretty hilarious too (I know because I have these games around and have enjoyed every drop of playability they give), but something like Gradius, or Thunder Force, no. It would just seem out of place in a more serious shooter, but could also work for a cute em up like Twinbee. Racing should never be considered for the most dire of shoot em ups, but hey the light hearted good clean fun ones, perhaps.
As for FPS's, those I don't consider shoot em ups, and I give give two shits less about whatsin/not in with them. Half Life 2 was the last one I cared about, and it didn't have racing, but you could ride around in a buggy (inputting a code to have a buggy appear anywhere, or just about anywhere is hella entertaining, but not racing).