Hi questioner-4805,
What's the version of Office that you get from your employer? Besides, please open one Office app, go to File > Account, check the version of Office installed on new computer, such as Microsoft 365 apps for enterprise, Office Home and Business 2019.
Office 2019, 2016, or 2013 could be installed once on a computer, to transfer Office from one computer to another one, please uninstall Office from the original computer first.
On the new computer, sign in to your My Account page and click Install. After you finish installing, you may be asked to activate Office over the phone.
For Microsoft 365, generally, you may install it on your devices and sign in to five at the same time. I would suggest you uninstall Office from your new computer, and reinstall it to check this issue. Here is an article about how to install Microsoft 365, "Download and install or reinstall Microsoft 365 or Office 2019 on a PC or Mac"