You indeed need a new licence for that second computer and it should be a full version licence. You can't use an upgrade licence since the computer didn't have another installed Windows before. You can't use an OEM (although, people still buy those separately even if it's not in accordance with Microsoft's EULA) because it doesn't come pre-installed with the PC. So, because of that, you need a full version.
It's up to you to respect or not Microsoft's EULA but I encourage you do so.