Absolutely yes, you can wear running shoes on the bike. For what I know from personal experience, experience from friends, and from discussion on forums and sites like this one, the shoes are one of the many things about bike equipment about what the answer to the question is "whatever suits you best". This is very common for riders to use a lot of very different shoes: sandals for grocery, regular shoes for commuting to work, clipless for trails or long distance, fast rides, and so on. You can wear what ever shoes you want to wear. As long as your bike has regular pedals I say go for it.