I had a similar problem recently.
My old computer had an education WORD as I worked for the school board and they installed it. I have now left the board and bought a new computer for similar reasons. (The old one was getting very laggy and the drive sometimes had to be interrogated more than once to get files.)
I bought a copy of 2016 Office. It loaded fine from MS but would not authenticate. Turns out it may have been a "Buy with computer only" copy. When MS Downloader first asks the input key it only checks for version before downloading the files. eg. If the key is a 2016 OFFICE key it will automatically download OFFICE2016. However when you are asked to input the key after downloading in order to authenticate MS checks the key against vendor type and previous use and may then disallow.
After about five weeks of back and forth my vendor offered me a copy of OFFICE 2019 which loaded just fine.
So if you original copy of 2003 was bought with your computer it almost certainly will not authenticate your new installation.
I have used a couple of other programs in which the key only opened the program in my computer and at least one of those has been running on my computers since about 1994 and runs just fine on W10. Sounds as though your problem is that your 2003 interrogates an MS server for authentication which gives them the opportunity to deny it each time you load it.
My problem with my current 2019 download is that it is a license for one particular computer only. If I lose the computer I lose the program. I am not actually buying the intellectual right to use indefinitely across platforms. (It was a lot cheaper though!)