InControl | input manager for Unity that tames the cross-platform | Model View Controller library

How can I share the View of the inherited controller without passing the view as a parameter? Am I doing something "mvc" wrong?

Yes, you are not using MVC as you are supposed to. So rather than answering how you can do this, I'll answer why you should not do this.

If you take a look at a definition for MVC, one of the main advantages is loose coupling. The idea is that you can change any Controller, Model or View and you know where the impact is (exactly where you are making the change). In this sense, having a base Controller or a base Model is a bad idea as you end up with tight coupling.

It seems that you are worried about code duplication, and that's perfectly fine. However, not all code duplication is bad duplication. There's useful duplication (like here, where you make it easy to change code later on) and useless duplication (like having some method twice when it could be one method and a parameter). At some point you will end up having so many different checks to see if it's one of the children or the other, that nobody will want to modify that code, not even you.
If you need to share business logic, the way to do this in MVC is by having another layer that your Controllers talk to, namely a services layer.

Also, there's almost never a good reason for doing stuff like this:

just add this