No you don't. You get the same license as you do on Steam, here's the license btw https://support.gog.com/hc/en-us/articles/16034990432541-GOG-User-Agreement-effective-from-17-February-2024?product=gog :

We give you and other GOG users the personal right (known legally as a 'license') to use GOG services and to download, access and/or stream (depending on the content) and use GOG content. This license is for your personal use. We can stop or suspend this license in some situations, which are explained later on.

Which is very similar to Steam. In both cases you can keep the files you've downloaded on your machine, and on most cases you can copy those files to a different machine and keep playing it. GOG has better marketing on this regard, but they're both very similar, neither enforces DRM nor forbids it entirely, although GOG does tend to be a bit stricter (but they still allow it) whereas steam is a bit looser but knowingly implemented a weak DRM and let's you know in the game page if the game has any stronger form of DRM.