A singularity is not required for something to be a black hole (in the literature, people sometimes talk about regular black holes). As you can check on more mathematical books on GR (such as Hawking & Ellis or Wald), the definition of a black hole is
Let $M$ be a strongly asymptotically predictable spacetime. If the set $B = M \setminus J^-(\mathscr{I}^+)$ is non-empty, it is said to be a black hole.
"Strongly asymptotically predictable" means the spacetime is sufficiently well-behaved at infinity so that $\mathscr{I}^+$ (the future null infinity, which you can roughly think of as all of the observers that go infinitely far away in infinite time, or where the light rays go to in infinite time) has some nice properties. $J^-(S)$ is the causal past of the set $S$, i.e., it is a generalization of the past light cone (including the cone's surface) for a set in a general curved spacetime.
A less mathematical way of stating the previous definition is
Let $M$ be a "well-behaved" spacetime. If there is a region $B$ in the spacetime which does not lie in the causal past of any observers that go infinitely far in infinite time, then $B$ is said to be a black hole.
The condition of the observers going to infinity in infinite time is to ensure they don't end up trapped in a black hole, for example. Notice that you have to ask all observers if each event in the spacetime is in their causal past. If some point is in the causal past of a single observer, then it is not in the black hole.
Notice then that this means that the definition of a black hole is global. You can't use only local properties to characterize whether black hole. In particular, you can't pinpoint locally where is the event horizon of a black hole.
With all this in mind, notice that the answer to your question is no. The metric alone is not enough to characterize the presence of a black hole, since it is a local construction. You need at least some other source of information. For example, you might need to clarify what is the topology of the spacetime you are considering. Notice that the Schwarzschild metric, for example, might or not lead to a black hole: depending on how you set up the spacetime, the metric might correspond to a black hole or to the exterior region of a star. It ends up depending on the ranges of the parameters that go in the metric (i.e., in the topology, a global property).
However, there is an interesting addition. If you happen to know you are dealing with a solution to the Einstein equations that has some nice initial properties (for example, it starts as a star and collapses to a singularity), then it is conjectured that no singularity can form without the formation of an event horizon. I.e., there is some belief that many problems of physical interest actually can't present a naked singularity. These are known as the Cosmic Censorship Conjectures. If you assume a conjecture of this form (and its hypothesis), then Penrose's theorem implies the presence of a black hole. Notice the Nobel committee did assume this when saying that "black hole formation is a robust prediction of the general theory of relativity". (Notice global information is already being assumed in the hypothesis to the conjectures and in whatever singularity theorem you use, and therefore this comment doesn't disagree with what I stated previously.)
Comments
Q: Does thermodynamic properties imply that a spacetime has a black hole?
A: No, thermodynamics does not imply a black hole: de Sitter spacetime also has thermodynamic properties, as I discussed in this answer, with some further comments in this other answer. I can't think of any thermodynamic properties that are specific to black holes, specially since their behavior is pretty much dictated by the usual laws of thermodynamics.
