I don't think remembering everyone's name is the point in that the professor doesn't say "in your life you'll meet many people" he says "in your careers". I think the point is realizing there are people who might be lower in the org chart who still serve a vital purpose and deserve your respect. Because even the CEO couldn't get his job done if there wasn't a cleaning lady.
So the very least you can do is to dedicate yourself to knowing the names of people who actually contribute to your success.
I respect almost every working person. That does not mean I should remember names of all of them. Even if I wanted -- I cannot -- there are more than a billion such people.
So the very least you can do is to dedicate yourself to knowing the names of people who actually contribute to your success.