I'm not aware of any one stat that captures everything a catcher can do, from game calling, general defense, pitch framing, blocking balls in the dirt, etc.
You'll sometimes see "catcher ERA", which is the ERA for all pitchers that throw to that catcher. It can be compared to games that that catcher doesn't catch. Lots of room for random variation there, for sure.
Maybe someday someone can solve the "how do we best value everything catchers do" problem in an all-inclusive way that somehow gets buy-in from the peanut gallery (tip, don't have two versions of it or have it fit into a "What's it good for?" classic song, like WAR).
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