@terziyski hey, thanks for your reply. If I understand correctly, the link you posted and the according Problem only applies to devices with the older green interface right? Because both of my devices have the newer blue one and I cannot find anything related. The only thing I can see is that the RE330 is set to extend the network and to apply the router settings automatically (the second option would be to make a WiFi network out of a cable-only network).

@terziyski So the MAC address of devices directly connected to the router is 1-1. It´s the same thing. The devices connected to the range extender do not appear in the router DHCP settings at all. Not as a virtual (slightly changed) MAC address, nor with their real one. They´re simply not there. 

The devices that do appear on both have different IP addresses. I guess that at some point they were connected to either of the 2 devices and are inactive on one of them now. 

I got one device e.g. that shows like this in my RE330 web UI:
sf8008 | 192.168.1.201 | 40-9C-A7-48-B2-5A | 5G

There is not anything even coming close to that MAC address (let alone the same one) in my router DHCP settings or Wireless statistics page.

@terziyski I tried your settings and now it is working as expected. After restarting the devices I was able to reserve an IP address for my TV receiver and also all other devices are now showing the same IP address on the router and the RE330. Thank you so much for your help :)