Characterizing OmiseGO as a DEX is perhaps not the most accurate assesment though. OmiseGO has the functionality built-in to exchange any token for any token (or indeed fiat money) which sounds a lot like a DEX certainly, but OMG merely provides the framework atop which could builda DEX. OmiseGO themselves are not building a DEX themselves, but another party could develop a DEX that utilizes the OmiseGO network.

What Omise, the parent company of OmiseGO, mainly is is a payments processor. They already handle online payments for thousands of merchants and customers in South East Asia in countries such as Thailand, Singapore, South-Korea and Japan. This existing network will be ported to the OmiseGO blockchain when it is ready. That means that there is an already existing network of users that will drive adoption forward.

As opposed to most projects that offer a 'this might be adopted', OmiseGO actually will implement it and therefore force adoption on their existing userbase. Every transaction on the OMG network will create fees, and those are distributed to the token holders. Since there is already an existing volume of transactions, it is almost guaranteed to create revenue for the token holders. The amount of revenue is still uncertain, as it depends on how the economy of the OMG network will evolve in terms of gas costs.

The DEX capabilities themselves are mainly to provide ease-of-use to any user. A participating merchant can opt to receive in any currency that he or she likes (for example dollars), and the customer can pay in any currency that he or she likes (for example ETH or OMG tokens). The conversion of one type to another requires a DEX functionality, and this is why the OMG blockchain has this feature. It is a means, but not it's primary selling point.

I do believe there will be many apps built on top of the OMG network, and some of these will be DEXes or Shapeshift-like services for sure.