Back in the day Flea was almost never seen without a Stingray, even though he rarely, if ever, recorded with one. But, now he has been spreading it around to the Modulus and vintage Jazz, so I don't think that bass is so associated with him anymore.
Almost every gig, someone in the audience asks if I know any a) Tool or b) Primus. But, while they are pretty well-known, at least among people know a bit about music, I doubt that any non-bass playing folks have heard of Wals or Carl Thompson.