I think you’re out of luck finding the serial number in a simple files backup. If you have a disk image of the laptop then maybe, but otherwise, no.
Your friend owns the product and you didn’t otherwise document what it was. The easiest thing would be contact your friend, but maybe that is no longer possible.
If you still had your laptop and it was a Windows computer, then there is a program you could have run to find the serial number, but without still having the computer with the software installed on it I don’t think you can find it, yourself. It may be in the registry or in an encrypted file. Adobe makes their licensing very complicated in an attempt to thwart pirates. If the serial number was just in a plaintext file it’d be too easy to hack.
If you actually owned the product and could prove so with a sales receipt then Adobe might provide a new serial number, but with a story like you’re telling, where you don’t even have the computer anymore and it’s not associated with the AdobeID you can login to and give Adobe it could just be a made up story to get free software since all evidence of your legitimately having it is conveniently missing. I’m not saying your story isn’t right, I’m just saying Adobe can’t merely give software away based on a good story, they require some evidence.
If you do still have a way to contact your friend there is an official transfer-of-license process that can be used to assign the serial number to your AdobeID INSTEAD OF theirs. If the situation is that they have one install and you had the other then your install would be illegitimate and Adobe wouldn’t have a positive attitude toward your wanting it back. If your install was used as a basis of a cheaper upgrade to CC by your friend and they don’t have need of it, anymore, that still might not be legit because Adobe gave your friend a discount based on ownership of the CS6 license.
If you friend is willing and able to transfer the license to you then this is the process they’d need to follow and Educational versions are not eligible if that’s also a consideration in this case:
https://helpx.adobe.com/x-productkb/policy-pricing/transfer-product-license.html