It can be extracted with a software tool (there are quite a few of them) but it's tied to the given hardware/OEM brand and can't be used for general purpose installations anyway. So, no they're not one and the same thing and in the case of Windows 8, 8.1, and 10 you can find COA stickers attached to laptops and desktops and the stickers probably don't even have a Product Key printed on them at all because it's stored in the BIOS/UEFI nowadays as part of the software licensing system in use. That is the actual license itself (more or less) which does mean you're legitimately licensed (give or take the situation) to have/install/use the given product.
Having a Product Key does not necessarily mean you're licensed to use the OS which is what the Certificate of Authenticity aka the COA is for. A Windows Product Key is just that: the actual 25 digit alphanumeric sequence written like AAAAA-BBBBB-CCCCC-DDDDD-EEEEE - that is simply used to activate when necessary since Windows 10 now uses a different scheme if you upgraded to it as part of the free upgrade path offer - there's still an actual Product Key if you purchase it in retail channels however but it's only needed one time to generate the original activation hash which now gets stored on Microsoft's servers for later use.
