Hey there @famcoll!
Yes, the tracking may still take place via sites that utilize Google services - but not as much has it does on phones. However, the router gives you the option to use the MAC address of each computer separately. With Linux, you can basically randomize your MAC address via the respective connection in the Network Settings. 
The MAC address (AKA Media Access Control) is essentially a form of serial number of your network card (ethernet or WiFi). It is used for device identification and is most used as identification for phones. Google doesn’t really use it much - if at all - on static/desktop systems, as they know it can be spoofed.