Microsoft's Web API MVC technology is typically used for external components to interact with the system - not generally a requirement for a standard MVC Web Application.
With that said, I'm not perfectly clear on the architecture. A few points of note:
- jQuery AJAX is a perfectly valid (and usually preferred) method to retrieve information for tab pages - it provides a SPA (Single Page Application) "feel" to the site and generally improves performance all around. This does not mean that they're using a Web API
- MVC is a framework used for many web applications, including Microsoft's Web API projects as well as ASP.NET MVC Web projects. The use of MVC doesn't mean that they're using a Web API.
- A RESTful approach isn't just for Web APIs. Indeed, many find it a cleaner approach when using regular MVC Web Applications, as it tends to be more semantic to what actions are actually being performed (GET to get a view, POST to post data, DELETE, etc.) There's no real reason not to use REST verbs (which are actually just HTTP verbs, but called "REST" when we use them in a certain way). The use of HTTP verbs doesn't mean that we're using a Web API.
To conclude, The MVC Web API framework is it's own technology that's similar to MVC Web Applications, but more geared towards working with non-visual requests and responses, instead tailored to programmatic interfacing.
If this is indeed a Web API being used and not a case of MVC practices that you happen to be unfamiliar with, then yes, I think it's a good question to raise (at least from a technical standpoint - politically, maybe not, but we can't answer that for you).
A typical setup is to have multiple projects, one of which is a Web Application, which makes use of shared project(s) for domain/business classes and data persistence. Additionally, a Web API project is often used to provide access to the system for external components, but this is a separate "presentation layer" project from the aforementioned Web Application.
There may be cases where a Web API application is written as the core interface between the internal system and the rest of the outside world, where the MVC Web Application then interacts with the Web API, but this is a corner case that should only be done with specific reason, in my opinion (unless I misread, this seems to be the case you're stating?)