I am observing this as well, 0.99.126-rc (WebKit 618.1.2) and macOS Sonoma 14.2 Beta (23C5030f).

For what it's worth, I have observed a "bug" in other app icons in Stage Manager where, even though an app has a custom .icns file applied, the "original" icon shows up in the Stage Manager interface. I bring this up because it seems plausible to me that there is some out-of-band icon being referenced for Stage Manager that could be affecting webapps as well.