His point is that no matter how much you think you are an expert in a particular field, there is always someone better than you.
Except for the one guy.
So never call yourself an expert and never let anyone else call you one either.
Oh, then I think he's made a mistake. It doesn't matter that somewhere, someone knows more entirely or just on a point. If you follow your professor's notion no one would be anything because qualification of any sort is based on a criteria and how our skill-set and knowledge relates. Why should expert be the unreasonable cut-off? Why not the initial designation, like lawyer or physician? Because those are labels that note a particular level of education in a field. Not much different from further education and expertise.
I know all sorts of experts in all sorts of fields. And I'm one in a few. It's only a way of recognizing a degree of familiarity and knowledge that rises to a level where the holder's voice is a measurably more informed and therefore generally more reliable summation of a particular.
If you do let your ego get the better of you. Be prepared to get egg on your face.
To me the mistake is the assumption that it's about ego. It shouldn't be, though if someone is proud of having gone to and through specialization in any particular and his acumen attests to the impact, that's his business. Won't make him winsome to a jury, past a point.