Bought new computer-purchased Photoshop 13 in 2023 and want to reinstall it on new computer. Don't want the installer(have it) can I reinstall 13 on this one??

I doubt either Photoshop Elements 13 released in 2014 or Photoshop CS6 released in 2012 would have been for sale in 2023

Photoshop 13 would be Photoshop CS6 which might be called »antiquated«, but please check the System Requirements yourself:

https://helpx.adobe.com/creative-suite/kb/cs6-system-requirements.html 

Another problem would be activating the license. 

Did you mean »Photoshop Elements 13«? 

If so please post on the corresponding Forum: 

https://community.adobe.com/photoshop-elements-714