When it comes to a noun + noun compound, the former is usually in its singular form. But why do you say "people management", not "person management"? If this is about managing a group of people, not a single person, then why you say "animal management", not "animals management"?
I think this comes from animals being seen as something you have a quantity of (think "bucket of sand" vs "bucket of sands") whereas people like to be seen as individuals, and not a collective mass of person.
If you are a "fun" kind of manager though, with just one person that you manage, you could say that you are in the business of "person management".