How do I get drivers for my motherboard (Z590 aorus elite AX) for Ubuntu

You will not need to do this in the future versions of Ubuntu when support for this Driver propagates. Hopefully not long now...

Common problem for Ubuntu 20.04 since year 2020

Just got it working personally now, but google directed me to your post.

Assuming your version of the Gigabyte Z950 Aorus Elite AX has the same 2.5GB Ethernet Chip. Then the below should work.

1. Download Linux Driver direct from Realtek website. Below link works now, but google if it doesn't. https://www.realtek.com/en/component/zoo/category/network-interface-controllers-10-100-1000m-gigabit-ethernet-pci-express-software I selected 2.5G Ethernet LINUX driver r8125 for kernel up to 5.6 released 2021/05/28 version 9.005.06
2. Download and extract downloaded tar archive
3. Open terminal and install build essentials..
4. sudo apt update && sudo apt install build-essential --yes
5. In extracted directory type
6. sudo ./autorun.sh

If the above steps work fine, and Ethernet cable connected, nothing obvious forgotten, then your Ethernet connection should be up immediately without restart needed