They'll recognize it as the browser they don't have any maybe should get because it's now positively associated with cool new features like this. :)

I think what helps is that there's two (or more) parties to a file transfer: the sender and the recipient. Someone who uses Firefox might start using Send, and then the recipient(s) finds out that they can use it too. And if they're using Send, the might start to consider using Firefox, or to create a Firefox account first.