There are two kinds of mass. Inertial mass is the amount by which an object resists acceleration when subject to a force. It's also the amount of energy an object has when it's at rest. On the other hand, gravitational mass is the "gravitational charge", the strength of the interaction between and object and the gravitational field.
Conceptually, it's possible to imagine a universe in which these two types of mass have different values. Maybe an object resists acceleration very strongly, but doesn't actually get attracted by gravitational fields that much. Or maybe you could have an object that produces a strong gravitational field, but doesn't have very much energy at rest.
However, this would be in violation of general relativity. General relativity assumes (and experiments so far have shown) that the gravitational mass of an object is equal to as its inertial mass. This is known as the equivalence principle. Regardless, the fundamental fact remains that an object's energy at rest and the way that it interacts with gravity are technically two different quantities.
The graviton, in this way, is similar to the photon. It mediates the gravitational field, and the "charge" that describes its interaction with particles is the gravitational mass.
The Standard Model works perfectly well without gravitons because the mass that it cares about is the inertial mass, which is (mostly) the result of interactions with the Higgs boson. The gravitational mass, the "charge" with respect to the gravitational field, isn't really relevant without a gravitational field, after all.
That said, there's not much of a reason to expect the graviton to be all that similar to the photon. They are very different types of particles, and the equations describing their behavior are, as a result, also completely different (specifically, the photon is spin-1, while the graviton is spin-2).