You don't mention which Ubuntu version you are using, but I'm guessing Ubuntu 22.04 since that release doesn't include FUSE by default. See this answer on Ask Ubuntu. I tested with the KeePassXC AppImage on WSL on both Ubuntu 20.04 and 22.04. It works fine on 20.04, but I get the same error as you on 22.04. To quote the entire error for searchability:
dlopen(): error loading libfuse.so.2
AppImages require FUSE to run.
You might still be able to extract the contents of this AppImage
if you run it with the --appimage-extract option.
for more information
Again, this isn't a WSL issue -- You'd see the same thing on any installation of Ubuntu 22.04.
The solution is straightforward:
sudo apt install libfuse2
However, on WSL you may find that you need additional dependencies for graphical apps, since the WSL Ubuntu distribution is based on Ubuntu Server and doesn't include graphical libraries by default.
For instance, for KeePassXC, there are a number of graphical dependencies in the AppImage that just aren't available with Ubuntu Server.
I'm honestly not even sure what all of the dependencies are, since I tried to install them piecemeal without success. However, if you:
sudo apt install xterm
... then it will also come with all of the needed graphical libraries for (at least) KeePassXC (and probably others).
However, there are almost certainly AppImages that have other dependencies, such as a desktop environment (e.g. Gnome or KDE).