Both gravity and EM fields have energy which is what couples to the gravitational field. Neither of the fields has mass, though.

> If all matter reacted to EM fields the same way it would to gravitation, would that make EM fields no force either? Or put another way, gravitation act universally on all particles, while EM fields do not. That necessarily has consequences when it comes to relativity. However it seems odd to argue that general relativity would exclude gravitation from being a force. If it acted only on a subset of particles, it would likely be in the same position as EM fields, and suddenly become a force again?

This is very well thought. Indeed, the equivalence principle, the fact that gravity couples to everything in exactly the same way (and that includes gravity itself as per the previous clarification) lurks behind our ability to reinterpret gravity in a geometric fashion. After all, if something didn't interact with gravity in the same way as everything else we could establish an experiment to differentiate if a spaceship is accelerating or stationary under a gravitational field (see Einstein's mental experiment) by measuring how that thing behaves. And that same fact would stop us from interpreting gravity as curvature of spacetime itself.

To your last point, speaking of forces is probably antiquated anyway, although still in use partly for historical reasons partly abuse of terminology. Preferably we should use the term "interactions", after all some of the "forces" do not result in push or pull as we usually understand a force in Newtonian mechanics but in things like color change. Others, like the gravitational "force" can be expressed entirely as spacetime geometry. But discussing semantics is quite pointless so as long as everyone understands in what way the term "force" is an abuse of terminology it's OK to keep using it.