Given that the President of the United States went on a yelling spree recently about Bezos' ownership, I'd say most people who can distinguish the Washington Post from the Washington Times (owned by Sun Myung Moon's religion and absurdly right-wing) also know who owns the Post.

(Which isn't to say everyone knows that either—a lot of people seem to take the Times equally seriously—but solving the problem of bad-faith media existing is another one entirely, and quite probably the answer is it shouldn't be solved at all.)