Settings up new "servers" is trivially easy and can be done without payment. You can join any number of servers with same account and retain separation (i.e. other users can not see a list of servers you are on). You can change your nickname on server-by-server basis. "Everyone" is already on Discord and by everyone here I mean every gaming community and many tech communities.
For TeamSpeak, Ventrilo, and Mumble (which were the big 3 in my youth at least) you have to have different clients, you have to pay for servers, and the chat support is very rudimentary and bad. Channels aren't as easy to create and admin. Lastly they don't support inlined links/images or emojis. Which makes them just as user friendly as IRC.
Discord took what was good about Slack's UI and added on top good VoIP. It might not be perfect but it quite literally covers most what anyone could want and it scales really well. To beat it you must have 100% feature coverage + another killed feature. First part probably isn't that hard, but the second is. There needs to be a reason to install yet another communication program on your PC/tablet/phone because all of the communities wont just uproot and leave to a new platform over night.