Black hole solutions to theories with no matter fields The Schwarzschild black hole is described by a mass M but comes from the Einstein's field equations with the stress-energy tensor set to zero. If our theory does not include matter fields - say in the Einstein-Hilbert action - then how can we interpret the mass M? What is the black hole made of if there is no matter fields? Is it the case that graviton vacuum fluctuations collapse into a black hole?
 A: From the point of view of classical GR, it is "made of" gravitational field -- there is a level where this question is akin to asking what electromagnetic waves are "made of" when there are no charges present.  (and yes, it is possible to have some particular "geon" solutions of Einstein's equations where you start with gravitational waves, and have a black hole at late times -- gravitational fields couple to themselves, so nonlinear effects like this that aren't possible in E&M are possible in GR)
You seem to be "what is this M in the schwarzschild solution, and what does it have to do with mass", then?  Well, there are several notions that we can use to generate an idea of mass, and match it to the $M$ parameter:

*

*Physical black holes came from a collapse event.  Thus, you can use something like the Oppenheimer-snyder solution, go and add up all of the mass that falls into it, and that's the "mass of the black hole"

*More rigorously, if you use Hamiltonian approaches to general relativity, in asymptotically flat spacetimes, you can use the symmetries of the "sphere at infinity" to define Noether currents that are interpretable as the "energy of the spacetime inside the sphere".  This gives you a notion of $M$, most commonly, the ADM or Bondi masses, depending on which "limit to infinity" you take

*And, of course, you can simply follow the original GR authors, and look at the motion of geodesic orbits, match them to the Newtonian results, and simply say "hey, this has to be $M$"

I'm sure that this list isn't exhaustive.
