Just because an institution is privately owned does not mean it can't have an agenda which is strongly tilted to the Left on a wide variety of social issues and causes, even if that tilt does not include the abolishment of Capitalism. Everything is privately owned in this society, after all.
As to CNN, while I used to laugh off FOX watchers claiming CNN was ridiculously biased and far left the last year has changed my mind. On social justice issues, particularly those involving the causes of anti-racism, homophobia, trans-rights and feminism they most definitely DO have an agenda which is quite far over on the Left of the spectrum. That's not to say FOX isn't far over on the Right, of course.
And the real problem with both is not their ideological tilt but the profoundly dishonest ways in which they misinform people with slanted news in order to convince them they're right.