Product keys may also be called CD or software keys. When programs were sold on CDs the key got called a CD key. Some people may refer to it as a serial number also.
If you are moving your software to a new machine or had to reinstall the software sometimes you will have to enter the product key again to activate the software, so you can use it. Keep in mind some software only allows the key used on one copy at a time. Usually it recorded itself back at the software company. Some software lets you use more than one copy. However, you may have to call the company to get them to release the activation no longer used so you can activate on new machine. Adobe and Microsoft, I have found make this painless.
Belarc Advisor is the best program for getting most product keys. It is available at belarc.com and has been around a long time. It will produce a list of all software on your machine and keys associated with them
However, the limitation with it is that it will show a key for Microsoft Office but that is not the real key but a hash of it. Other programs will find that same key. Recently I discovered if you did buy directly form Microsoft or registered with them it is possible in most cases to get the key. You will need to know email address you used to register the product. The site is https://support.microsoft.com/en-us/help/4000641/microsoft-store-find-your-downloads-product-keys
If it is possible the best solution is keeping the product ID cards and keep the others all written down. If Microsoft Office came pre-installed on your machine (not the trial version) then usually they put sticker on the case with the number just like they do for Windows.