client-server API architecture

is it architecturally correct to "merge" the client and server API's into a single assembly?

I wouldn't do that.

Having interfaces and necessary information in common (by which I'm assuming communication protocol and data transfer objects) does not mean that clients and server are strongly coupled, unless they share the code that does the logic or makes the functionality happen.

Having clients and server in different assemblies adds a layer of separation that forces you to keep the functionality decoupled and makes you think deeper about where certain code should reside, inside the client or inside the server. If you accidentally couple server and client too much, assemblies will no longer compile independently.

But if you deploy them together into one assembly, it will be easy to end up with too much coupled code without knowing it, simply because everything sits together and compiles together. And when you need to add another client or separate an existing one further, you will have a harder time decoupling them at that point, than if you kept things separated from the beginning.

There are good things that come out from keeping code loosely coupled, separating responsibilities and separating concerns, which you can't really substitute with "but now my build creates one dll file instead of four".