I think the equals sign actually works perfectly well for this purpose. For starters, "enharmonic" itself is really a short way to say "enharmonically equivalent", so from a language perspective, = makes a lot of sense. In any context where it is important to note that two things are enharmonically equivalent, it will be obvious that the normal distinctions about enharmonicity not being entirely the same are implied. Plus, = is such a commonly understood shorthand that its meaning is immediately obvious to even beginners.
As an alternative, I think the similar (~) sign from geometry could also work really well, and it probably comes closer to the exact meaning of enharmonic. The congruent symbol takes this even further, but isn't localised on many keyboards.
The main argument I have against other symbols, like the ones mentioned in other answers from mathematics or logic, is that they aren't nearly as accessible as an equals sign. Just about everyone in the world knows what an equals sign generally represents, but lamentably the same cannot be said about the congruent symbol, for example.
I've seen scores sneak their way from C major to D♭ major, and when their G♯ note gets tied to A♭, the composers (or editors) sometimes write "G♯=A♭" above the notes.
If you're okay with shorthanding the enharmonic relationship with a symbol, you have to be okay with letting the technicalities of enharmonic relationships be implicit. Otherwise, you may as well just write it out.