Hannity is not a journalist, even according to Fox, himself or Wikipedia:
Sean Patrick Hannity (born December 30, 1961) is an American talk show host and conservative political commentator. [...]
Jim Rutenberg commented in August 2016 that Hannity is "not only Mr. Trump's biggest media booster; he also veers into the role of adviser," citing sources who said Hannity spent months offering suggestions to Trump and his campaign on strategy and messaging. Hannity responded to the report by saying, "I'm not hiding the fact that I want Donald Trump to be the next President of the United States. ... I never claimed to be a journalist."
So Vox saying something about "any other journalist" is fairly misleading here, even if true.
Basically Hannity is a talk-show host on Fox... even though he does most of the talking himself, I think. Fox (theoretically) breaks its talk-show segments apart from their news segments. This distinction is probably lost on many, but:
At Fox News, opinion is king – not news
And this is hardly a new thing, Fox has been like that since forever basically:
The motto of the network is "Fair & Balanced", but in many respects it is anything but. "Fox News is not really a news network, it's a commentary network. Its news output is a small island in a vast sea of very conservative commentary," said Mike Hoyt, editor of the Columbia Journalism Review whose March issue features a cover story exploring the Fox News phenomenon.
Per discussions in comments below, the Vox claim could perhaps be reinterpreted more broadly as saying some like "any other prime-time TV talking head (other than Hannity) who did something similar to what Hannity did, got fired for it". I haven't investigated that angle in terms of data/facts [a lot], but it could be a[nother] reasonable way to answer this question, I think. (This formulation implies that the "talking head" is an employee of the network; there are a lot of guest/invited "talking heads" on TV talk shows who aren't network employees, and the claim would/could not apply to them.)
In general, talk-show persons are held to lower political-bias standards than journalists though as this story relates how one journalist fired for displaying a MAGA hat reinvented himself as a conservative talk show host:
A former Rochester TV reporter fired last year for wearing a "Make America Great Again" hat at last year's Trump rally in Rochester has a new gig and says now, "I think it just exposed the truth about the state of journalism in this country."
Jim Bunner is now host of his own conservative internet radio podcast on Central City Radio. And far from being chastened by the MAGA episode, the one-time TV multimedia reporter for the NBC affiliate KTTC-TV is using it to market his show and reach potential listeners.
Bunner had a moment in the national limelight in October 2018 as a result of the MAGA hat episode. Bunner was captured by a Star Tribune photographer wearing the hat while covering the Trump rally at Mayo Civic Center. The perplexed photographer tweeted the photo with the caption, "Whyyy?"
Bunner was fired by KTTC-TV the next day. The news director at the time said the station does not allow staff members to cover stories while wearing apparel from political campaigns. The story created national headlines. [...]
Bunner never gave an interview in the immediate aftermath of his dismissal to explain how the hat wound up on his head. Later, Bunner said a man attending the rally put it on him, likely as a joke, assuming that Bunner would take it off. But Bunner didn't.
Bunner said he kept wearing the hat not as a declaration of allegiance to Trump but as way of developing trust and rapport with Trump supporters. He said that when he arrived at the rally — but before having the hat thrust on his head — he was largely treated with scorn and derision by the Trump crowd. [...]
Where before it showered him with scorn, Bunner said he actually had people approaching him and showing a willingness to be filmed by him. One woman tapped him on shoulder and invited him to interview all of her kids, "which is rare because people are usually camera-shy," he said. [...]
Bunner said it was never his intention for anybody but Trump supporters to see him wearing the hat. But then the Star Tribune "paparazzo" took a picture of him "without me knowing it and without my permission and then it just went viral all over the internet."
(The fact that Bunner now hosts a conservative talk show puts the prior explanation (merely building trust with those filmed) in a bit of a different light though.)
Anyway, given that story, my guess (insofar) is that the broader claim that other pundits (as opposed to journalists) would be fired for (something like) what Hannity did is more likely to be false, whereas the claim that journalists would get fired (especially if they did it "on the job") seems more likely to be true, but I haven't done anything like a comprehensive survey on this. (Of course, we're talking about US journalists here, in some authoritarian countries, the journalists are much more likely to be mouthpieces/endorsers of whoever is in power.)