I don't think Firefox is immune to this, just that because its architecture is different, Anthropic didn't bother coding a bridge for it (given its market share).
The main issue here is that Anthropic violated one of the most important (implicit) tenets of applications in a computer: don't touch other people's shit. Claude.app modified Brave (and others) configuration, adding an extension without user consent. An extension that, by the way, gives full control of the browser to Claude, including reading the DOM for browser tabs unrelated to Claude (for example, the one where you just entered your credit card details).