Photoshop Elements 13 auf neuem Computer

You can get download links here:

Direct Download Links for Adobe Photoshop/Premiere Elements 13 | ProDesignTools

Unter meinem Adobe Account kann ich das Programm unter "Registrierte Produkte" sehen, jedoch lässt es sich nicht auf meinem PC neu installieren oder aktivieren (ich bekomme nur Elements 2020 auf der Website vorgeschlagen).

I think you have met the same problem I had with PSE2021: I got a wrong installation file for 2020.

See that discussion:

https://community.adobe.com/t5/download-install/photoshop-elements-2021-order-re-installs-2020/td-p/11516238?page=1

Try to download from the link I gave in that discussion.

Edit:

I re-read the title of your post, and I think I missed the fact that it's the PSE13 version you wanted to reinstal ?