Networking – check which devices are connected to the router I’m connected to through a range extender (Booster)

When using a repeater the repeater usually will have the same SSID as your WiFi AP. However, it will always have a different BSSID (think of it as of the MAC address of a network interface).

So, in order to tell to which device you're connected, you have to find a way to display the BSSID you're (currently) connected to. While I was able to find a command for windows which will show you all the known BSSIDs of a SSID:

netsh wlan show network mode=BSSID

I was not able to display the BSSID which is currently used (eventually someone here is aware of such a built-in command). However, I found a tool which will show you the available BSSIDs and make it possible for you to connect to a specific BSSID: checkout NetSetMan.

Once installed and started, go to "Tools"-->"NSM WiFi Management". You'll see all SSIDs listed and if a SSID has multiple BSSIDs - as it will if it's using a repeater - you can look at those BSSIDs if you hover over the SSID with your mouse.

If you're on Android you can use Apps like "WiFi Analyzer" which will also present you with the BSSID.

Are there any methods, specific plates or materials which can boost the signal?

You're likely to succeed only to degrade the signal or its strength - fine-tuning radio transmission requires special equipment and intimate knowledge (as in schematics and calculations) of the actual gear used.

There is no "magic" tin-foil trick that can be used. If your device has a connector for an external antenna that is your only option - but may well turn out to be of little gain, depending on the antenna you choose.

The actual construction of the walls may be relevant; i.e. if there is a lot of iron bars inside concrete walls, or if there is sun-protection film on sheets of glass... - these would attenuate any radio signals.

Note: there are WiFi devices that operate at 5GHz - as these are quite new, you might get less disturbance (few neighbours have them) on such a device - the reach might not be as long though; higher base frequency also implies less reach.