This is such an important point. I'm living with a generic "group" feature in a SaaS app and it's a nightmare. It's been used by numerous clients for totally different things. These are the weeds that will choke your garden.
We have the concept of 'tags' in our app, which are used by our users to delineate anything from company departments, social groups, locations, age bands etc. - basically anything they see fit. It saves us creating a mess of drop downs to try and cater for all edge cases, and they feel the freedom to think of them however they like.