"Left" and "Right" hasn't meant anything in America for a very long time except "Democrat" and "Republican" and neither party has ever been ideologically coherent. What happened in recent years after the civil rights era opened up a more competitive South is that a massive sorting process pushed out from both the crossover members but this wasn't ideological it was simply happenstance that more "conservatives" were in the Republicans and vice versa. All kinds of natural systems promoted this, I don't think it was really too deliberate as much as like attracts like. The original Progressives were, for example, in both parties at the time. They wanted extensive state control, more "experts" with unlimited power, more voting (but not to remove the "experts" of course), and oh yeah a shit load of racism and eugenics to get rid of those disgusting classes holding back society from being perfected by the experts. Were they leftist or rightist? Doesn't even make sense to the mainstream view. Arguably the entire "left" and "right" European context makes no sense in American politics because America has never had a king and the "elite" has always been transitory rather than a permanent social class of privilege established by tradition. Canada and Australia are not too different in this regard but they always had connections to the home country that could establish the "right" faction through similarities. (And I would be remiss if I didn't point out that the "left" faction in France after the Revolution quickly pushed the liberals over to the "right" as enemies undermining the revolution and established a dictatorship focused on striking down heretics betraying the revolution in every walk of life.)
The easiest way to see this American spectrum incoherence is foreign policy where the party that controls the Presidency is generally interventionist while the party out of power becomes skeptical about it, all the ideological and theoretical supports are twisted and redirected to support this rather than stay consistent. Because of this there are all kinds of political science tomes (like Tony should have read in grad school) trying to explain why foreign policy somehow doesn't count instead of everyone admitting that the traditionally taught political spectrum is the thing that doesn't make any sense and never has unless you really squint at some single element. (I'd contend not even really in Europe except for a specific period, it's a figment of classifying parliamentary majorities and minorities as somehow displaying linear coherence. Germany's earliest democratic parties, and still arguably to this day, make a complete mess of any linear spectrum.)
Tony is a "leftist" in the American sense because he's chasing cultural trends associated with Democrats in mostly blind loyalty. He has absolutely no clue why he's personally gone from supporting a party that touted "ending welfare as we know it", supporting tax cuts, putting 300K cops on the streets and wanting great state spying capabilities, bombing Iraq and that "you can't say you love your country but not your government", saying marriage is between one man and one woman to opposing the Patriot Act, opposing bombing Iraq, opposing tax cuts, opposing deficits, demanding government transparency, supporting standing up to your government to opposing transparency, bombing Iraq, supporting tax cuts, declaring political opposition terrorists opposed to democracy, randomly deciding gay marriage is good after all to supporting abolishing the police, supporting the largest government welfare expansion ever, saying deficits don't matter, supporting greater state surveillance and I'm picking just a few here to illustrate the incoherence. Tony probably, like most people, thinks he's been consistent the entire time and maybe gotten more "leftist" consistently (and the opposite on all this for blindly loyal Republicans) when maybe the one thing 2000 Tony, 2006 Tony, 2012 Tony, 2018 Tony and 2022 Tony would absolutely agree on and maybe actually shifted more to the "left" on over those years is abortion. It's a cultural phenomenon not an ideological one.
Tony, much like Brianna Wu, feigns to the activist left and hints at "true loyalty" at their more coherent socialist viewpoints but has to hue to the Democratic mainstream because that's the only place for the social climbing he wants which makes him a slave to the political winds. If you could steal some of Rowling's black magic and somehow associate the Democrats with the "right" in everyone's brain while changing literally nothing about their positions all these people would have to switch up their lingo but wouldn't change anything else. (I'd further argue you wouldn't need to redefine the Republicans as the "left" in your spell as the inherent two-party confrontation system and need for sports like winners/losers plus the mandatory good/evil framing would quickly associate them there by default.) On troonery he breaks from this because he actually considers it personally important and therefore relevant enough for him to follow the crowd there rather than the crowd he really wants to climb in. Though I also have to note Tony's very long held belief that becoming the top expert in some area is what will allow him to climb in the group he wants to, so his desire to become the top political troon can probably be seen in this framework rather than a true allegiance even to troonery circles.