The answer was that these questions, when posed as philosophical dilemmas, were bullshit.

I've always interpreted them as prompts to introspection. The sages of old gave advice along the lines of "Know thyself", which naturally leads one to ask, "Who am I?" I can say, "I am a mathematician", which is fine when I need to list my profession but that line of questioning seeks to get past such superficial details. The purpose of the advice is to strip away the facades we put up to interact with others and to unearth the lies we tell ourselves. You can find the similar sentiments dressed up in the language of psychology and cognitive science on this very website. The questions are vague because you're compressing a lot of information into a simple, easy-to-remember form.

As you've pointed out, in everyday usage context will dictate the appropriate response. The serious sages wouldn't expect you to articulate a response because what they really want is to prompt some serious reflection. That leaves us with:

So when a young philosopher follows up your response with, “no really, who are you?”, you should respond with asking, “what in particular would you like to know?”

If you want to disengage from that conversation, the old "I am I" works. Or you can be graceful and say, "I'm still trying to figure that out." The sort of young philosopher who asks these questions is unlikely to actually want or expect a "proper" answer.