Thank you for these Florentia and Chez.
The person who used the question thought it could be an equivalent for: 'what are you like?' that you ask when you want to know about someone's character (shy, friendly...).
What's your personalityis a possible question in context. And the context I have in mind is where "personality" has a predetermined meaning. For instance:
They say most people are either introverted or extroverted. What's your personality?
"Jungian personality sorters" assign people to one of sixteen distinct personality types, along axes of introverted (I) or extroverted (E), Sensing (S) or Intuitive (N), Thinking (T) or Feeling (F), and Judging (J) or Perceiving (P).* So, what's your personality?