I think you’re getting downvoted because this doesn’t make much sense.
[…] it is only on Linux that […]
And which other platform is Flatpak available on? As far as I know it’s just available on Linux. You are probably comparing two entirely different things.
A password manager can’t communi[cate] with a Browser […]
Flatpak is a utility for software distribution and sandboxing. And that is what it does. Sandboxes are a means of isolating software from the rest of the system and each other. By using Flatpak to install software you deliberately choose not to choose the version of the browser that would have been tied into the rest of the system… And now it isn’t tied into the system…
I’m sorry if I sound a bit negative, but this is how it is. Lots of people use Flatpak because it is popular, but it is the wrong tool for many users. Some don’t know how it works and why they should or shouldn’t use it. I always recommend you install packages from your distro first, unless you have a valid reason not to… And then you need to pay attention to which tool you use and how to use it. Flatpak has means of connecting software and the feature to give more permissions to software or lessen the sandboxing. But since they do isolate software, it doesn’t come with that as a default. Also there are other tools to distribute software you could choose. And I think this specific situation with the Firefox Flatpak has gotten better. I think they’re slowly addressing stuff like that but it’s probably not something they focus on. I don’t really know the specifics of this situation, I installed Firefox or LibreWolf and the password manager from the package repository. It works and the themes also work.
(Edit: I’m not sure but imagine the ‘flatpak run’ issue might be a security measure. It might require less correctness to just download something, but it needs to be precise once you execute something on your system. This would be analoguous to how some browsers allow you to download executables from the internet, but require you to confirm you want to run them in a separate dialog that pops up. Or the additional ‘chmod +x’ on Linux. You often don’t want to run random stuff by mistake. That could be probably implemented differently if this is the case.)