Not really. Although I live in the DC area and subscribe to the Post, and every article that has so much of a whiff of conflict of interest includes a parenthentical aside that Bezos owns the Post. We're used to it.
This isn't likely to do financial damage such that Bezos would care, and both Bezos and WaPo "win" if WaPo looks "impartial" enough to criticize Bezos - WaPo gets to show that off for journalistic credibility and Bezos gets to show off that the media org he owns is totally willing to speak truth to power, which makes him One Of The Good Billionaires.
This isn't likely to do financial damage such that Bezos would care, and both Bezos and WaPo "win" if WaPo looks "impartial" enough to criticize Bezos - WaPo gets to show that off for journalistic credibility and Bezos gets to show off that the media org he owns is totally willing to speak truth to power, which makes him One Of The Good Billionaires.