As an interviewer, I'll first gripe that I probably wouldn't ask this question unless I wanted to see how the candidate responds to massive ambiguity. The ability to actually answer this question aptly is not something I'd expect from most people. Especially not technical ones.
But - taking it from the interviewer's perspective - he's right. Most people that have made it through a screening process will tell you that they are smart, they are dedicated (or work hard) and they have the skills and experience to get things done. So it's hard to distinguish one candidate from another when we get into this.
What I find useful in this case is the idea of what unique qualities do you bring to a team and do the job? People are not cookie cutters - they have likes/dislikes, personal points of pride, particular ways of thinking - all of which make them unique, and when people are put together in a good team, they manage to compliment each other. As an interviewer, I'm taking this into account - I don't want a team that are carbon copies of each other, nor do I want a team that is so wildly ecletic that they can't form a cohesive whole.
I put this into the realm of "personal branding" - I didn't even realize there was jargon for it until recently. Alot of the advice in this area can sound really flaky - but the thing that sticks out is that who you are is more than either specific activities on the job, or being a geenrally ideal employee - is how you do what you do and what you put the most focus in. Some of these aren't going to change, you are who you are, so the trick is to "brand" them as an asset.
Here's my example -
I'm the summer camp counselor of digital security. I have all the training and experience in security that you see on my resume, but the real deal is that I like having people get along and having them work together to get big things accomplished. I'm never going to be that guy sitting in a cave saying "no" as loudly as possible - I'll be in there trying to figure out how to get the job done, while preventing risk adequately. It can mean that I come off as more permissive than most security gurus, but it produces a net win, because people don't circumvent me - they go to me for help because they trust me.
That's alot more than just "I've got a solid technical expertise" or "I'm a great team leader" - you get a sense of who I am and what I'm like to work with. I'm taking a bit of a risk - if you wanted a hard core, yelling and screaming security guy who was going to bang on the table an shriek about the dangers of the internet - then you won't hire me. But if you are worried that your ops folks won't trust that sort of person and you need a good collaborator, you're probably drooling at the idea of having me on your team.
At the same time, if I didn't like the people stuff, and I'd spin a different tale about being ... the Sherlock Holms of digital security or something similar.
In the end, though, my proposed strategy means committing to something. Talking about strengths/weaknesses, it can be easy to avoid pigeon holing yourself into an area where you don't come off well. Talking about "what makes you special" really honestly, can mean that they go "yep, that's special, but not what we want". The trick is being confident enough that you've sold a good story, and if they don't like who you honestly are, then better to find it out before you quit your current job.