Meaning of ‘what’s-that-under-your-cloak innuendo’

The passage below is from A History of the Index by Dennis Duncan.

We have found ourselves right back with the very earliest books to be printed in English, and already anxieties about the index – its use and abuse; the dangers of overdependence – are apparent. We have looked forwards to our own time, touching on a parallel with twenty-first-century fears about Google’s effect on our capacity for deep reading. Now, let us round off an anxious chapter by looking backwards to the earliest iteration of the same worry. Plato’s Phaedrus is a dialogue between Socrates and his young friend Phaedrus as they stroll outside the city walls of Athens before resting under a plane tree. (The pair, it seems, are more than just friends – the opening exchanges are brimfull of flirting and what’s-that-under-your-cloak innuendo.) Like much of Plato’s writing, the

Phaedrus contains a number of satirical attacks on the literary world of fourth-century Athens. When Socrates first bumps into Phaedrus, the latter is on his way from spending the morning with the great orator Lysias, listening to him give a speech on the subject of love. Socrates asks his friend to repeat the speech to him as they walk along, to which Phaedrus replies in shock: ‘What are you saying, my dear Socrates? Do you suppose that I, who am a mere ordinary man, can tell from memory, in a way that is worthy of Lysias, what he, the cleverest writer of our day, composed at his leisure and took a long time for?’ Socrates teases Phaedrus – I bet you asked to hear it twice, then borrowed the script so you could memorize it – and this, it turns out is precisely what happened. Phaedrus produces Lysias’ written speech from under his cloak, and the pair settle themselves down in the shade so that Phaedrus can read it out. At the end of the reading, Socrates is deeply moved. He declares the reading ‘miraculous’ and himself ‘quite overcome’. But it was not Lysias’ words that produced such an emotional response; rather it was Phaedrus’ performance of them: ‘This is due to you, Phaedrus, because as I looked at you, I saw that you were delighted by the speech as you read. So, thinking that you know more than I about such matters, I followed in your train and joined you in the divine frenzy.’

I want to ask a few questions for the underlined part of this passage.

First, the implicit meaning of ‘what’s-that-under-your-cloak innuendo’.

It’s not difficult to get the literal meaning of it. It’s not so hard to understand ‘something that’s under your cloak’. But it modifies ‘innuendo’, which here seems to mean double entendre, that is sexual innuendo. (Am I right?) Then, ‘what’s-that-under-your-cloak’ implicitly hints that it is a male genital. (Am I right?)

Last, ‘fourth-century Athens’ seems anachronistic. According to Wiki, Socrates lived from c.?470 BC to c.?399 BC. Then it’s not ‘fourth-century Athens’ but ‘fourth-century BC Athens’ that is the right timing.

Thanks in advance.