What does the server "supply" to you and your neighbour?
A server is simply a device (or a process) that provides some sort of a resource to users - there are file servers, print servers, database servers, terminal servers, just to name a few - and these have nothing to do with WiFi which is what we call the network "physical" layer - so to speak a communication medium.
I have a server in my home network, I can turn it off and it won't prevent my family members from accessing the internet using WiFi - it will prevent them from being able to retrieve files stored on the server, but they can still browse the worldwide web, watch Netflix, use Facebook and so on.
Unless you can tell us how the piece fit together and work, there's very little we can do other than make a SWAG, and that's not going to be much help.
Maybe you should be having this discussion with the service provider.