Could I make a signal of the devices connected?

Some routers may get computer names using Windows networking (and maybe Apple Zeroconfig), but not all connected devices do that, if file/printer sharing is not enabled or non-Windows devices. A 2Wire DSL gateway (that I used for years before its wireless got too weak) allowed configuration of names for connected devices and would resolve those names in its local DNS. But that is probably rare for gateways or routers to do that.

I did not even remember that the Motorola DSL gateway that I am using now has a way to add alias names for connected devices until I just looked at its config now.

The names are read from each connecting device. If you carefully label your machines, you will see those names in the router list. It is done from each machine, not from the router. In many cases just knowing which MAC addresses are your machines is just as valid. Each computer lan adapter will have a specific MAC address which you can read from the network settings on each machine.