Yes, and it has been done before.

As others noted, it's handy to have just two states, "on" and "off" (actually "mostly on" and "mostly off") because it works well with digital circuit design.

But it can be done, and in some cases (like flash memory) we sometimes use 3 or 4 states. On very rare occasions, even more. This vastly improves some things like storage density, but hurts other things such as complexity and signal to noise ratio.

One of the very good reasons to not use more than two states for most digital circuit design is power. When designing circuits where transistors are "mostly on" or "mostly off", it is possible to arrange things such that there isn't much current flow most of the time. When you add intermediate states to the transistors, it is much harder to avoid current flow and hence power consumption. This is a big deal in circuits that have millions or billions of transistors, even if each individual current path consumes little power.

You could introduce more "states" without them being represented by how much individual transistors are turned on. But that makes the circuits a lot larger, so you lose most of the potential benefit.

Of course, another reason now is legacy software. Everything is written for binary logic. While that's a problem that would be possible to get around, it would be painful. You'd need a darn good reason to do so.