Seconds out for Grudge Match 2: the ageing bull versus the old pretender. It’s a trash-talking, fist-pumping, shadow-boxing spectacular featuring two rich alpha-males who hate each other. Initially, these men appear to be worlds apart. But as they lumberingly circle the ring – trading insults, playing to the gallery – it becomes harder to tell which is which.

Here is the story so far. In 2016, Robert De Niro makes a video in which he calls Donald Trump a punk, a pig, a dog and a bozo and says that what he would really like to do is punch the guy in the face. On Sunday night he barrels on stage at the Tony award ceremony, shouts “Fuck Trump!” as his intro and then shakes his fists in the air as though he already has. Trump duly hits back, labelling De Niro “a very low IQ individual [who] received to [sic] many shots to the head” and advising him to “Wake up, Punchy!” So let’s score the first round to De Niro and the last one to Trump – although he gets docked a point for misspelling the word “too”.

I’m not sure how much this brawling adds to the US’s political discourse, but it’s bizarrely revealing about both De Niro and Trump. I’m not even convinced this qualifies as the culture wars, exactly. Far safer to view it as a spat between two Gotham millionaires with bulging property portfolios and a shared interest in New York gangster movies. Their language is identical; their mode of attack is the same. In what sense does this battle highlight the differences between them? If anything, it shows how much they have in common.

De Niro, for instance, refers to Trump as a “bullshit artist” and “a con”, implying that the man is a bad actor who somehow struck gold. In attempting to map out the president’s pedigree, pundits typically reach for fictional references. They liken him to Lonesome Rhodes, the demagogue folk singer in Elia Kazan’s 1957 film A Face in the Crowd, or Greg Stillson, the deranged politician from Stephen King’s The Dead Zone (later played on screen by Martin Sheen). All of which is fair enough – and yet Rhodes and Stillson were creatures of the American heartland. They didn’t have Trump’s peculiarly New York aesthetic. They didn’t possess his blend of peacockish flamboyance and vicious invective; the sense that he has plucked his personal style from a low-rent mafia flick. “I like him as an actor,” Trump once said of De Niro. And here, perhaps, we arrive at the man’s real reference point.

Look back over the actor’s most lauded performances. Every now and again, you catch a glimmer of Trump. He is there in embryo as Johnny Boy in Mean Streets, a swaggering player who likes to humiliate his creditors. He is there as Jake LaMotta in Raging Bull, particularly in his impotent final scenes as a bloated nightclub compere. And he is nestled in the DNA of hand-flapping Rupert Pupkin, the thick-skinned mediocrity out of King of Comedy. Reviewing King of Comedy at the time of its 1982 release, Pauline Kael dismissed it as “an empty movie [which] seems to teeter between jokiness and hate”. Substitute the word “movie” for “man” and she might be describing the current occupant of the Oval Office.

Or how about Taxi Driver, in which a maverick cabbie embarks on a mission to clean up depressed, crime-infested, mid-70s New York? Taxi Driver, you will recall, famously ended with Travis Bickle storming an East Village brothel. Trump, at around the same time, earned his spurs by purchasing the crumbling wreck of the Commodore Hotel – then, reportedly, a hang-out for sex workers. Both men went in with the odds stacked against them. Both, in their way, came out hailed as heroes.

“The irony at the end is that [Travis] is back driving a cab, celebrated,” De Niro once pointed out. “That’s relevant in some weird way today, too, with people like Donald Trump, who shouldn’t even be where he is, God help us.”

None of which, of course, is the actor’s fault. He has rustled up some brilliant performances and we should be thankful for that; his responsibility ends there. It is not up to De Niro to police how these roles influence members of the general public, whether that is would-be presidential assassin John Hinckley Jr or the president himself. His hands are clean; his conscience clear. He is free to rail against Trump just as much as he wants.

All the same, it is a curious thing – the parallel course these two men have charted off-screen. De Niro, let’s not forget, is a New York real estate mogul. He owns hotels and restaurants and a large swathe of Tribeca. His son, Raphael, is the developer behind a record-breaking $120m listing on Manhattan’s Upper East Side. And Trump, for his part, once nursed movie ambitions. He appeared in Zoolander, Two Weeks Notice and Home Alone 2, like Hitler with his oils or Charlie Manson with his music. If Hollywood had only seen fit to embrace him more fully, the world might have been spared the man’s horrendous plan b.

Apparently Trump’s all-time favourite De Niro film is Goodfellas, Martin Scorsese’s swirling, teeming portrait of New York’s gangster scene. Peer hard enough and you can find glimmers here, too. He is not Jimmy “the Gent” Conway, who is altogether too coiled and watchful. Nor is he young buck Henry Hill, who falls in with the mob and then rats them out. But wait, over on the sidelines stands Morrie the wig-man. Morrie is verbose and pathetic, a fast-talking huckster who longs to run with the wise-guys. He says things such as “I’ve been good to you, you’ve been good to me” and “Hey, come on, who loves you more than I do?”. It’s clear from the start that he’s heading for a bad end.

In the final act of Goodfellas, Morrie is ushered into an alley and literally stabbed in the back. This seems a natural fate for the Morries of this world, men who play-act the tough guy and get in over their heads. It’s just that real-life has a knack of veering from the script. Sometimes, incredibly, Morrie manages to survive and prosper. The comical copycat starts to outpace his idols; the duff tribute act takes on a life of its own. And when that happens, the fictional boundaries break down. The ultimate gangster performance is no longer confined to the screen. It risks being played out for the masses upon the world stage.

At the Tony awards at Radio City Music Hall, De Niro was ostensibly booked to introduce his old friend Bruce Springsteen. He had a pre-prepared script that had been rehearsed in advance. But under the lights, with the cameras rolling, he decided it was better to wing it instead. Who knows whether his subsequent outburst was prompted by righteous rage, or revulsion, or a kind of horrified recognition? But in the moment it reminded me of another famous De Niro improvisation, way back in the 70s, on the set of Taxi Driver. It’s the one that takes place inside Travis’s lonely apartment. The man at the glass; the side-arm in his coat-sleeve. “Saw you coming, shit-heel,” he sneers. “You talking to me?” Poor Travis, it transpires, has created a monster. The scumbag looking back is his own mirror image.