For the moment, it's not officially on the Amazon Appstore. You can try installing it manually via the "Downloader" application by searching for the Android APK.
Be careful, though: if you install the mobile version (for phones), navigation with the Fire Stick remote may be difficult (you may need a virtual mouse app). In the meantime, the easiest thing to do is to use the stick's built-in web browser (Silk) to visit the Galaxus TV website.

I have the same question at the moment. Unfortunately I haven't found an APK for Galaxus TV yet. Does anyone have a new hint?

It is theoretically possible, but often the apps do not work correctly on the stick or are extremely difficult to use. The installation is also correspondingly time-consuming. If you still want to try, here is a quick guide: "To download and install the APK, you must first activate the developer options on your Fire TV and switch on "Apps of unknown origin" and "ADB debugging". Then use the downloader app to search the Internet for the desired APK file, for example on sites such as AFTVhacks or APKMirror, and download and install it on your device." There is also an extended app store that offers you all kinds of apps for download, but even there you often encounter difficulties. Note: Apkx files do not work.