The note sort of answers your question. If there is no other attestation of Socrates' writing (there isn't), then nobody to our knowledge would have thought Socrates was a writer

I don't know why Dobbins thinks this is such a mystery. For that matter, I don't know who Dobbins is, but LPh records him as having published a monograph (I assume his dissertation) on Epictetus, though he only has three publications to his name, two on Virgil, as well as the Penguin translation. He's not an active scholar as far as I can tell, or at any department.

In any case, aside from what Dobbins thinks, this passage has attracted a fair amount of notice from scholars, but not because it's a mystery. The passage is interesting because of Epictetus' use of Socrates as a philosophical exemplar guiding his own Stoic thought. Epictetus does not say "Socrates was a writer" and does not call him "prolific," he says Σωκράτης οὐκ ἔγραφεν; καὶ τίς τοσαῦτα; "Didn't Socrates write? And who wrote so much!" Epictetus is either anachronistically imposing his own practice on Socrates or referring to imagined or unprepared writings. The distinction is unimportant: Epictetus is not referring to published, or even existing writing. He's not saying that Socrates was an author. He says as much: the comment that no one wrote so much as Socrates is followed by the imagined protest ἀλλὰ πῶς; "But how could that be so?" He's referring to his own practice, which he mentions elsewhere in the Discourses, of using writing as an aid for self-examination, and to Socrates' own ἔλεγχος, his cross-examination of interlocutors according to his method. It's quite clear what's going on here, because Epictetus explains what he means later on. He says that what he means is that even when Socrates was alone he cross-examined himself, using writing as an aid. Epictetus did the same thing, and is using Socrates as a model for his work, even though Socrates did no such thing. The whole point of Epictetus' explaining this is in answer to his imagined ἀλλὰ πῶς; question, the implication clearly being that he's imagining an interlocutor protesting that Socrates didn't write anything, and responding by saying that he's not really talking about "real" writings.