On the GitHub Marketplace, there are both Actions and Apps. With a GitHub app, one is able to customize and and add logos with ease, and configure the icon of a bot-commentator. However, with actions, there is no way to customize this.
The only workaround I have found is to create a GitHub bot that listens to a GitHub action and then comments. This is really inconvenient, as GitHub actions are already so powerful and can do all the functionality I need, and I’d really just like to customize the branding on the action.