Unlike the mobile app, the desktop app continues to miss a built-in feature to select an alternative icon. This applies to the macOS desktop edition, unknown behavior on other desktop platforms.
When I backup …/Brave Browser.app/Contents/Resources/app.icns and overwrite with a custom icon, then the icon appears in some macOS places (Spotlight) but fails to alter the dock icon.
Relaunched Brave. No luck.
Killed the Dock process. No luck.
What’s wild is that Spotlight has caching bugs; Even after restoring the original app.icns, Spotlight continues to show the custom icon indefinitely. But the dock never does.
Various reports online recommend graphical, shell script, or even paid third party apps to change the icon. All of those methods are awkward in practice.
Contrast with the mobile app, which showcases a built-in option to tailor the icon among a pool of several alternative possibilities. On desktop, we just get brokenness.
A related issue was closed years ago by obsessive janitor bots without resolution: