Are you using Windows or macOS? As far as I can tell, there is not a permission request that I need to grant for it to work in other apps. The Grammarly desktop app allows you to block it from working in specific apps (but I have not blocked it in Obsidian).
The floating Grammarly button is how it is supposed to work. In macOS, it shows up next to whatever textbox I am writing in. Then I can select it and see the recommendations that it offers/click the underlined text.