If your computer gets an IP address from your ISP via DHCP, releasing the IP address may unregister the MAC address on your ISP's end. Here's how to try:
Go to a Command Prompt, type
ipconfig /release and press Enter. When you get the result showing an IP address of 0.0.0.0, unplug the Ethernet cable from your computer.
Plug the Ethernet cable into the second computer. Open a Command Prompt, type
ipconfig /renew and press Enter. If it works, you'll see a valid IP address appear. If it doesn't work, it will take a minute or so and eventually show a message that the request timed out.
If it does work, you'll have to remember to do the same release process before unplugging the Ethernet cable on each computer.
As others have mentioned, a router is a more convenient solution that will allow you to use multiple computers on the same connection simultaneously. If your ISP does unregister the MAC address when you release the IP address, then you can release it before connecting the router to avoid having to find the router's MAC cloning settings.