I have 2 computers, both came with Windows 10 Home licenses. For one of them, I've upgraded it to Pro using my employer's volume license.
Is it possible to downgrade the computer with the Pro license and do the upgrade in the other one with the same license? And if so, what would be the procedure?
Is it possible to...upgrade...the other one with the same license?
No. Volume upgrade licenses for the Windows OS merge the original license with the upgrade license, creating a single licence. It's a violation of the EULA to separate them once a system has been upgraded.
Your organization's Microsoft licensing administrator should have access to the full text of the EULA governing your volume licenses and be able to confirm this.
Note that I've not addressed whether it's technically possible to do this given doing so would violate the EULA and put your employer at risk of failing a Microsoft license audit.