I read through the entire thing, in French, and it references several other laws and European regulations. The browser part is only in those paragraphs of the bill, but those paragraphs are referenced throughout the bill so there might be more to it. The whole bill is 91 pages long.
It was adopted on 5 July, by the way...
The blog post from Mozilla does say, "France’s browser-based website blocking proposal will set a disastrous precedent for the open internet," so the precedent here is what they're worried about.