Thank you for posting, @smart_dynamism48a6, and thanks to @kglad, @John T Smith, @leo.r, and @Peru Bob for their input so far.
I reviewed the account details available to us and was unable to locate any registered Adobe software titles associated with the Adobe ID currently in use. Based on that, it appears you may be signed in with a different Adobe account than the one that was originally used to register Photoshop CS5.
Adobe automatically registered perpetual-license serial numbers to the purchasing account, so if you can locate and sign in with the correct Adobe ID, you should be able to view the registered product and access any available installation files associated with that account.
Once you've identified the correct Adobe account, please review the following document for information on downloading older perpetual-license products:
https://community.adobe.com/announcements-616/download-non-subscription-adobe-software-554002
I understand how frustrating this is, especially when you're trying to move to a new computer after your previous system failed. Hopefully, locating the original Adobe account will help you recover access to the registration information you need.
Please update this public discussion thread if you have any additional questions, @smart_dynamism48a6.