This makes no sense. Stored procedures are code - just code that executes within your DB, instead of on your app servers. I agree that it makes sense to keep your business logic all in one codebase, but there are times when stored procs are a very efficient solution to business-logic problems.
Overall I agree that this "DB API" solution is not entirely elegant, and might provide as many security holes as it does interesting debugging challenges.
However, there are some situations in which I could see this being incredibly useful. Combining this Nginx's JSON/API modules means some parts of your API could entirely bypass your app servers.
Let's say you have an API that shows the ten items most recently liked by a particular user. Typically, this request would hit Nginx, your app server, then your cache and/or DB. By using the "shared procs API" and some Nginx modules, you could provide the same data by letting Nginx query your cache/DB directly.