Unstated by Baier, but the other reasons Sullivan is considered "liberal" by Fox News is that he is opposed to torture and was a critic of the Bush 43 administration. Nevermind that Sullivan's criticism of President Bush 43 is a conservative critique.
Sullivan previously addressed his being identified as "liberal" by Forbes magazine and made an astute point in response. That he is considered liberal not because he favors gay rights so much as because of implicit prejudice on the part of fellow conservatives
The real truth is that many on the Republican right just read everything I write through an anti-gay prism, because their homophobia - benign or not-so-benign, conscious or unconscious - is so overwhelming it occludes any genuine assessment of a person's thoughts outside this fact. See how Forbes cannot even keep the word gay out of quote marks. Just imagine the same sentence with the word "Jewish" replacing the word gay. It tells you everything you need to know about the moral core of conservatism today. It's sad and will one day be seen as embarrassing.I don't find his second guess at why he's considered "liberal" (because he's critical of Christian nationalism) as convincing. While I'm sure that contributes, I think it's primarily his ardent opposition to torture that designates him as liberal to the Fox News crowd.
Authoritarians seem to have great difficulty wrapping their head around the notion that those designated as "terrorists," "enemy combatants" or "the worst of the worst" by those they consider leaders can be mistreated; or that they share the same fundamental human rights that the legitimacy of our form of democracy is grounded in recognizing (i.e. unalienable rights.)
Anyone who thinks otherwise falls out of the in-group category of Us (conservatives) and into the category of Them (liberals) in their Manichean world view.