Hi HamAndHands,
Unlike your Microsoft 365 product., my Microsoft Home and Business 2021 for Macs products are not subscription-based (e.g.,)where there are periodic (annual?) payments to use the software. My Office software are stand-alone, one-time purchases with no recurring subscription payments involved. That said, since approximately mid-January 2024, I am seeing a "Update License" link under the Word, Excel and PowerPoint "About" menu drop-down. Here's a sequential series of screen shots that illustrate what happens next ....
"Update License" link appears ....

Clicking on link opens pop-up window to "Sign in" to verify ....

This opens window to "Activate Office" ...

Don't have a work or school account and select my Personal account (which exists, is legitimate, and has my 2 licensed Office products ...

Clicking on the Personal Account selection immediately returns a pop-up stating "Couldn't update license Error code: 44"

I can log into my Microsoft account and see my 2 Office Home and Business 2021 products and download them to reinstall the software successfully. In an effort to resolve the Update license Error code: 44 issue, I have removed and reinstalled the software multiple times. I have also used the License Removal tool to delete the licenses and then reinstalled them. So far, nothing has worked, and I've spent many hours over many days without any signs of success. I am still able to use all of the Office for Mac applications without issue. Other Mac users have been experiencing the same Update License issue with error code 44. over the past week or two, and haven't been able to solve the problem.
You are the first person I've seen who's encountered the same "Update License" issue, but on a PC, rather than on a Mac. In my opinion, this is a cross-platform issue and is the result of a recent software update deployed by Microsoft. I do not anticipate that efforts by end-users will be able to resolve this Update License issue. Microsoft engineers will need to be engaged to solve this.