It is a limitation of a PWA. A native app (wrapper) will likely be coming though, and a custom Capacitor plugin could theoretically interface with the native API to switch out icons. Theoretically. No promises though :)
Quoting this comment by the dev
I think it’s acceptable, though I am working on my own … for now experimental … Lemmy client in C++/Qt/QML … maybe eventually for desktop and mobile devices alike :)
A desktop client would be a game changer for sure.