So what is this “New Left” I’ve mentioned a couple of times now? Well, here’s where the hippies come in.
In the 1950s and 60s, some thinkers began applying certain forms of left-wing thought to ideas other than class struggle. The horrors of Stalin’s regime had prompted a lot of rethinking among leftist groups outside the USSR, even among self-identified communists.  In the US and elsewhere, leftists started discussing issues of gender, race, and sexuality alongside (or even instead of) issues of the working class. The orthodox Marxist view of society, interpreting everything in terms of class struggle, didn’t seem that satisfying any more.
Feminism already existed, of course—the suffragist movement started in the late eighteenth century. This was the beginning of what’s called second-wave feminism, which shifted its attention from voting and property rights to issues like discrimination and harassment in the workplace, domestic abuse, the depiction of women in the arts and media, and (eventually) abortion. The civil rights movement formally began in 1955, and it too had a new focus: not just legal, on-paper equality for people of color, but practical, socio-economic equality as well. The forerunners of the modern queer rights movement also emerged, in the Mattachine Society. This is why the counterculture movement of the 60s, which formed largely around opposition to the Vietnam War (a pretty traditionally leftist cause), also frequently involved a radical rejection of accepted norms about race, sex, and religion.
The common thread here (to a leftist) is forms of social oppression or exclusion, forms which you can’t understand very well on a simplistic poor-versus-rich model and which aren’t always expressed directly in legislation.  This is where identity politics comes from. This is also one part of why there’s so much infighting on the Left. As Natalie Wynn eloquently put it,
Anyone with any experience in leftist circles knows that Marxists and identity politics activists are constantly at each other’s throats, because the Marxists accuse the activists of being bourgeois dogs who want more female CEOs of color and more disabled transgender drone pilots, while the activists accuse the Marxists of being a boys’ club of brocialists no more woke on gender and race issues than the average Jordan Peterson fan. Most often these accusations are correct, because everyone is problematic and I disown them all.
I think the reason Americans confuse the Left with liberalism is that it’s fairly easy to adopt one or two of these causes and still be a liberal—and that is exactly what the Democratic Party has done.  Against a backdrop of identity politics, making a black lesbian the president of the bank is a great way to look like you’re reforming the bank, without doing anything to risk your own paycheck or help the actual debtors.
 In fact, the derisive nickname “tankie” is used by leftists to this day, to designate people with Stalinist or totalitarian views. It originally referred to western communists who sided with the Soviets during the Hungarian Revolution of 1956, which the USSR ultimately put down via invasion.
 There are lots of ways different forms of oppression can overlap with each other; this is why intersectionality matters, even if the word sounds annoying.
 This is why the Left really hates Biden, and didn’t much like the Clintons either. Their actual policies are solidly neoliberal, even if they make more allowances for social programs than the GOP. (Plus the whole sexual harassment thing ain’t great.)