Welcome to the forums! 
Yeah, Realtek keep cranking out new variants of chipsets almost every day, and motherboard manufactures like them because they’re real cheap. For most of these, large parts of the driver code are almost identical, so the kernel devs started to aggregate all these in the r8169 kernel driver. Unfortunately, they take some time to keep up with all the model variants (the letters after the number)…
I assume the newer r8125, r8126 and r8127 chipsets will also end up in the r8169 driver—which is a little unfortunate regarding the naming.
In the meantime, you could try to get the latest Linux drivers from the Realtek download pages, they’re usually pretty current (but check the kernel revisions). You’d then have to install the correct one and blacklist the r8169 driver so it doesn’t try to take over.
We’re just having such a situation with the r8125bg, see Ethernet funktioniert nicht (German).
It helps reporting the exact model of motherboard and Realtek chipset to the kernel devs—it’s often only a rather small change they have to make, and we’d all benefit from the driver becoming in-tree, i.e. part of the kernel.
That said, I’m not so much a driver guy, there are some people here much more knowledgeable than I am in that respect.