How do I connect my Mac to my router via CAT6?

Once that adapter is installed, you typically have to click the Add [+] button under the list of interfaces in

System preferences > networks ...

... to select the interface to be added to the list of interfaces,

Then use the gear icon to "set service order" to drop it TOPMOST so it will be used first.

Then configure as expected.

Be sure to also set IPv6 to "link-local only".

This allows Apple Bonjour Discovery protocol to work quietly behind the scenes and find Printers and potential shared resources on your local Network, without messing anything else up.

What problem are you having?