Migration often fails to transfer all files necessary for Microsoft Office and other applications like Adobe CS. It's best to use an installer to get all the bits and pieces.
You need the CD Key not the Product ID.
The article to remove Office will identify the files used to license Office. You can copy these over. You still have to activate. The process should be automatic but often the servers are down and if you have to call they will want the CD Key. It's possible they will help if you have the Product ID.
Office for Mac 2011: Remove Office
The Product Key also called CD Key is on a yellow or bright orange sticker on the CD package. It will look like this: ABC4Z-Y9ABC-Y6MNO-K2ZZZ-P23EF
The Product ID is given during the Setup Assistant process. The Product ID will look like this: 11111-130-0001111-12345). You can find it under About for the app.
I suggest getting
1Password or similar app to save important info like your CD Keys.