Lionhearted girl

Not much better. So he asked, “Did you and the priestesses have a celebration?” “Yes, though the service was the main highlight.” “I see.” She angled her head, hair shining like molten metal. “Do you sing?” He blinked. It wasn’t every day that people took him by surprise, but . . . “Why do you ask?” “They call you shadowsinger. Is it because you sing?” “I am a Shadowsinger—it’s not a title that someone just made up.” She shrugged again, irreverently. Az narrowed his eyes, studying her. “Do you, though?” she pressed. “Sing?” Azriel couldn’t help his soft chuckle. “Yes.” She opened her mouth to ask more, but he didn’t feel like explaining. Or demonstrating, since that was surely what she’d ask next. So Az jerked his chin to the sword dangling from her hand. “Try cutting the ribbon again.”

After their initial greeting, Azriel and Gwyn share a brief vulnerable moment in which he remembers the incident in Sangravah, Gwyn reads this in his expression and then attempts to end the conversation (assumably because she has no interest in being seen in that way). This sets the tone for the exchange moving forward. Azriel responds by keeping the conversation going, deliberately lightening the atmosphere by teasing her gently about kicking him out. What follows is some banter and casual conversation which they both choose to keep going. Azriel is not particularly smooth about it, having to attempt multiple times to give a good answer to her (pretty low-stakes) question about the party. It's a sensitive subject for him so he finally redirects by asking her about her celebration. Then we get the exchange about his singing. When he cuts her off before she can ask anymore, many interpret this as some sort of signal that he wants to shut her up but this is verfiably not what happens. In fact, instead, he redirects the conversation one more time, this time to the ribbon, instead of ending it (which he could have done). This is the second time he changes the subject in an effort to keep the conversation going without it touching on something that is just a little too sensitive for him to talk about. But that is not actually the interesting part.

See, when Azriel asks Gwyn about her celebration, Gwyn tells him about the service. As readers, we know that music and singing is a really integral part of the priestesses' services. Singing in particular is something that helped Gwyn heal from her trauma and thus holds a special significance in her life. So when she thinks of the service, she thinks of singing. And that is why she is prompted to ask about his own singing here. However, Azriel doesn't appear to be aware of the importance of singing and music for the priestesses, which is why the question takes him by surprise. He doesn't intuitively connect the service with singing so from his perspective she is probing at something personal about him, seemingly out of the blue. This checks out because this moment is literally the first time we ever hear about his singing. It's something he's kept well enough hidden that neither Feyre nor Nesta ever saw him do it. Based on this, we can infer that singing is something quite personal to him, likely not something he is comfortable just showing off. And the question came with no warning, leaving him a little exposed.

I find his reaction here hilarious actually. Gwyn opens her mouth to say something and he is convinced that she will ask him to sing "that was surely what she'd ask next" - he is under the impression that he has a pretty good read on her. Azriel is pretty terrible at reading Gwyn throughout this scene actually. He:

isn't warned by his shadows that she is in the training ring before he arrives

is convinced that he fooled her with his lie about needing to retrieve something ("The lie was smooth and cool, as he knew his face was." he is confident that he is unreadable here, he knows his face doesn't betray any emotion. Gwyn just smiles and only in the next exchange she calls him out on his lie by correctly inferring he needed time alone)

thinks she must be cold when she isn't

is surprised by the questions about singing

is puzzled at her not showing any sort of deference to his shadowsinger title

chuckles involuntarily at her pressing him about the question

This is just inference but I suspect Gwyn wasn't going to ask him to sing at all. Singing matters to her too so I think it's much more likely that she would've said something along the lines of "Actually, I/we sing too during the service!" bringing the conversation back to where it had started: the solstice celebration. But Azriel clams up so she never gets to create this bridge between them. To me, this seems like authorial choice: the obvious connection between them regarding singing and music is being revealed to the reader, but not (entirely) to the characters. And it's served its purpose very well, Gwyn and Azriel hearing each other sing or singing together is something that is highly anticipated by the readership. This is where that seed was sown and it was done intentionally.

It is also telling that he redirects the conversation into the student-instructor dynamic that he is far more comfortable in. It is evidence how much the question about his singing had knocked him from his usual confidence: he had to retreat into a social register that he feels in control in.