For the moment, at least, bitcoin is recognized (legally) as more of a product or commodity. This means that the legalities of running an exchange at present are a bit less like the stock exchange or a banking institution and much more like a specialized version of eBay. eBay matches sellers of various products with people willing to give them money for said products, which is exactly what a bitcoin exchange does also - match sellers of bitcoins (legally a product, not a currency or stock) with people willing to pay (in USD/EUR/etc.) for that product.

Now given enough time the legalities might catch up and decide once and for all what bitcoin truly is (currency? commodity? product?) and legalities may change, but it'll likely be at least a few years before that happens. Legal wheels turn slowly.