from what I read, the description on the card is misleading … either you have Fast Ethernet (100) or Gigabit (1000). And ethtool shows that you have 100 only. I assume the switch port it connects to, is orange instead of green.
my laptop has the same Realtek card (RTL8101) and ethtool shows only 100Mb. It’s a hardware issue, your device works at 100Mb, it cannot go to 1000. I connected my laptop to the gigabit switch, I still get 100Mb hence the orange color. Other devices on the same switch are green because they are capable of 1000Mb. Hope this helps.
TBH, it shouldn’t. I have the same chipset in my laptop and verified the same in both Windows and Linux. Realtek stuff is pretty generic and, in my experience, performs equally well, regardless of the OS.
If you really want to check, open the property sheet for your network card in Device Manager. Click the Advanced tab and select Speed & Duplex in the property field. Click the dropdown under Value and you’ll see all the options for your hardware (be sure not to change it). In my case, it matched the info given by ethtool.