Most images of black holes are illustrations. Here’s what our telescopes capture

... where the vast majority of that something's mass and charge are within the event horizon. To keep Birkhoff happy, and to avoid confusion with event horizons that pop up where a black hole clearly isn't, maybe add that in a small region of spacetime mass and charge inside an event horizon is a black hole if after stationarization electromagnetic and gravitational perturbations in the Schwarzschild metric are small compared to those in e.g. the Minkowski or Robertson-Walker metrics.

I'm tempted to go the other direction: a physically reasonable arrangement of stress-energy can source a black-hole-like metric and that a black hole metric (i.e., an exact solution for a mathematical black hole, like Kerr-Newman) can usefully approximate such that there is a good match between the geodesics structure around a mathematical black hole and the behaviour of observed matter in the vicinity of a black hole candidate.

A practical probe of the null geodesics structure around a BH candidate is surface emissions: we routinely detect them directly and through analyses of radiative efficiency for neutron stars and white dwarfs (whether or not these compact objects are accreting), and a detection of surface emission from a compact BH candidate would rule it out as a source of a BH metric. If all BH candidates have surface emissions, then BH metrics are a poor choice of modelling tool. However candidates which survive this probe and other tests of near-horizon geodesics structure might as well be called black holes, even if we have reasons to hope (or even doubt) that its mass and charge within the apparent horizon is concentrated in the singularity point.

You'll note here that I'm taking a "quacks like a duck" view of astrophysical black holes; I'd go even further and want to be agnostic about event horizon vs trapping surface and so forth. Around BH-like objects, arranged by increasing ease of observation, there will be a predictable (set of) ISCO(s) below which free-falling trajectories always decay or plunge inward (or outward if retrograde); there will be very strong gravitational lensing; there will be characteristic outflows from Penrose-like mechanisms; and a characteristic efficiency in conversion of accretion matter into radiation compared to matter accreting onto non-BH objects of similar mass. These all depend on whether the exterior region near the candidate object is like that of an exact BH solution, and when you have all of the above, and no evidence to the contrary, it is pretty safe to assume there is something very similar to an event horizon dividing the near-exterior region from the non-exterior region(s).

This has always confused me. Does a photon have mass? I've always thought the answer is no and so I don't understand why even light can't escape from a black hole.

No, they actually are holes. We don’t need analogies or metaphors to describe black holes. Their mathematical properties are quite complicated to define and we have to reason about what happens around or inside them using figurative thought experiments, but their action does correspond to our intuitive sense of what a hole is.

In particular, a black hole consists of an event horizon, around which various stable and unstable orbits are possible, and within which is a theoretical singularity. The event horizon perfectly corresponds to the mathematical concept of a hole, which is a pure abstraction of our common sense of a hole, not a metaphor. A hole is a lack of points in a dimensional space, which means everything inside the event horizon is as much a hole as a manhole in a street is a hole (information theoretically speaking, the inside of a black hole is nothing). To say a black hole is a hole is not a cognitive metaphor, because space around the event horizon actually does curve down to something that is a physical hole. Instead, here are two examples of cognitive metaphors:

1. A donut is a 3-dimensional space with a topological hole in the middle of it. Suppose I define the mathematical properties of taste and equip it as the only sense you have for investigation. Then the hole of a donut tastes like nothing because it is nothing, in the same sense that we can know nothing about the inside of a black hole because it is nothing.

2. Roll a quarter at an angle down a spherical curvature with a hole in the center. The quarter will gradually descend down the curvature, with each revolution about the center happening faster and faster. Finally, it will simply drop it. This is analogous to deteriorating your orbit around a black hole, until you enter various unstable orbits and finally fall into the event horizon.

Obviously these cognitive metaphors, while instructive, are imperfect. For instance, we can see a quarter drop into the hole, but we’d never actually see a shuttle fall into the event horizon. On the other hand, the event horizon is a hole in the same sense that a manhole is a hole. Comparing it to a manhole is another cognitive metaphor; calling it a hole isn’t, because it is one.