That depends on the software used to control the WiFi-logins. And maybe on the kind of trouble caused by a device.
If you want to know this for a particular system where you log in, ask the owner. As far as my experience with such systems goes, it’s usually not an account you have to enter, but some code that’s valid for a certain time after the first use.
It depends on the owner if he makes a note of what code they gave to whom. It depends on the system if it reports the code used by the device that gives troubles.
Almost any router can show you clients that are connected to it (most commonly in form of MAC addresses).
The rest depends on the setup. If there’s any sort of IDS installed, or a packet analyzer, admin should be able to figure out what machine (and, therefore, which client) is doing what.