I am glad you asked this.
I went researching which kernel for that driver and... it goes back a ways. Pre kernel 5 - you are using 6.8
And that hints to me that there may be another side to this coin.
Is your driver blacklisted?
We can check.
grep -r "blacklist" /etc/modprobe.d
We can scan through the list to see if r8168 is on there.
If it is, we can remove it from /etc/modprobe.d/blacklist.conf by opening that file As Root
sudo -i
nautilus
Navigate to the file in the above path and right click to open with Text Editor then removing the line blacklist r8168 EDIT: (Below, we show three full lines about r8168 to remove)
Saving the file after.
After saving the file, you must run
sudo update-initramfs -u
Then try modprobe for the 6.8 kernel included drivers
sudo modprobe r8168