As far as I understand, he's only conscious for short intervals, I'm guessing the priority wasn't finding out his name but rather checking for any discomfort and speeding up his recovery. Also, Korean names aren't as straightforward as Western ones like 'Anna' or 'Julian'—they can be literally translated into other meanings, like 'sunrise,' so there are more names to choose from i suppose. And since he can only hear, as was mentioned previously, asking him to point at the characters is also not an option.

The name thing is kind of stupid, they could have just show/read to him a list of names till he recognised his own