That is GOG Galaxy. So, “Does GOG Galaxy support Linux?” is a different question than “Does GOG support Linux?”, and the answer to that question is “no”. But if you really need a local launcher software like that for some reason, there is Heroic Launcher, which definitely does support Linux. Though realistically, you don’t need GOG Galaxy, Heroic Launcher, or any similar software to play the games. GOG is just a storefront to purchase DRM-free games. You can simply download those game install files using a web browser and install/run them normally.
But gog makes a launcher and library tool. But not for Linux.
That is GOG Galaxy. So, “Does GOG Galaxy support Linux?” is a different question than “Does GOG support Linux?”, and the answer to that question is “no”. But if you really need a local launcher software like that for some reason, there is Heroic Launcher, which definitely does support Linux. Though realistically, you don’t need GOG Galaxy, Heroic Launcher, or any similar software to play the games. GOG is just a storefront to purchase DRM-free games. You can simply download those game install files using a web browser and install/run them normally.