“The thrill of victory, the agony of defeat,” went the tagline for the long-running TV show The Wide World of Sports.

We’re all familiar with those rollercoaster emotions whether we follow professional football or dabble in sandlot softball.

But in recent weeks, a surprising new emotion keeps cropping up in the sports world: warmth for, and unity with, our fellow human beings. In our divisive and polarized world, that could not be more welcome.

Consider the visit to Boston of the Scottish national team earlier this month as part of the World Cup competition. Through some sort of ineffable serendipity, the kilted athletes, their gregarious fans and the stereotypically stuffy Bostonians embraced each other.

“What happened at Fenway Park on June 14 was something none of us will ever forget,” wrote Sam Kennedy, the president of the Boston Red Sox, in a letter to the leadership of Scotland’s team. “We knew the Tartan Army was coming. We did not fully understand what that meant until we saw it.”

Kennedy went on to describe how hundreds of Scotland supporters gathered at the foot of a statue of Scottish poet Robert Burns and marched the distance to the ballpark to the sound of bagpipes as Bostonians stood cheering by. He called it “one of the most moving things we have witnessed at Fenway Park in a very long time”.

The embrace of the two cultures spilled over into the city, as the NBC affiliate TV station in Boston put it: “They’ve marched through Boston, attended a Red Sox game at Fenway Park, played the bagpipes, tried out the city’s viral cop slide, put traffic cones on the heads of statues like Samuel Adams, made friends with the locals, cheered as Mayor Michelle Wu signed documents kicking off a sister city partnership with Glasgow and drank beer. A lot of beer.”

And the Boston Globe enthused that the Tartan Army’s “joy and awe are healing us”.

“Amazing, really,” a Boston-area friend texted me about the generosity of spirit shared between the visiting Scots and the locals. “So glad I experienced it!”

Similarly, in New York City – where Gotham residents are far more likely to mind their own business than chat up strangers – the New York Knicks championship brought all kinds of people together. Huge watch parties in every borough and an overflowing victory parade created a feeling of joyful unity in bodegas, workplaces and even on the subway.

“In this transformed city, previously forbidding strangers are transformed into fellow fans,” the New York Times wrote in their morning newsletter a few days after the final win over the San Antonio Spurs, the Knicks’ first such NBA championship in more than five decades.

The communal feeling is precious, with the Knicks providing “a rare pathway to intimacy”, wrote Melissa Kirsch, with reference to what British anthropologist Victor Turner termed “communitas”, a feeling that moves us off the usual script and into each other’s hearts.

Of course, this is a temporary state that’s sure to fade, but perhaps something can remain, some sense that we’re all in this messy, unpredictable life together and can recognize each other’s humanity.

That’s what happened in Buffalo this past hockey season before a playoff matchup between the NHL’s Buffalo Sabres and the Boston Bruins during the pre-game singing of the Canadian national anthem. (It is traditionally sung at all home Sabres games in the border city, along with the American anthem).

When featured singer Cami Clune’s microphone cut out, the mostly American crowd took over. They came to the rescue with a rousing and word-perfect version of O Canada, with its closing words: “O Canada, we stand on guard for thee.” And then the arena exploded into joyful cheers.

The episode, which soon went viral, was a balm to Canadians who have had their very national sovereignty threatened by the US president.

“I cried,” one Canadian, Linda Arcand, told a reporter from Buffalo’s WKBW television. “I couldn’t believe they were doing that. It makes me teary now.”

Whether featuring bagpipes, faulty mics or rowdy watch parties, these sports moments can provide a different “thrill of victory”.

And although fleeting, they have the enduring power to inspire. For a few moments or a few days, divisions crumble, replaced by the beauty of kinship.

• Margaret Sullivan is a Guardian US columnist writing on media, politics and culture