Conservatives don't mind being called conservatives. Even folks who aren't all that conservative, but who identify with that political alignment, are proud to call themselves conservatives. "Right-wing" is not a dirty word to them. They don't shy away from it. They've been calling themselves conservatives for well over a hundred years.
Their opponents, on the other hand, slip around like weasels, reinventing and renaming themselves every so often. Social reformers of the mid-19th century were Victorian liberals. Then the modern progressive movement started around 1890, drawing largely from the socialist ideas of Marx and Engels. Many different socialist factions formed, openly and proudly calling themselves socialists. Then the Great Depression happened, and modern liberalism formed in reaction to it. FDR introduced the New Deal, stealing the thunder of the socialist movement in the United States.
In the 1940s, the Democratic Party--and the liberal movement itself--became strongly anti-communist and purged leftists from their ranks. This left the Left as fringe outsiders without representation. We at that point had two pro-capitalist, pro-business, pro-elitist parties in America--the Democrats and the Republicans. Until the Civil Rights Era, there was little distinction between the two parties.
The Red Scare continued through the 1950s, but the Left still existed, mostly in the shadows. In the 1960s, the counterculture of rebellious youth embraced the Left, even to the point of supporting Maoism, and pushed back against "the Establishment," which included both the Republican Party and the Democratic Party. They criticized Democrats for not doing enough to fight capitalism and racism. America saw the rise of militant leftist groups like the Black Panther Party and the Weather Underground Organization.
But as the leftist hippies grew up, got jobs, and ran for public office, they voted Democrat and called themselves liberals. They still held the same political beliefs of equal opportunity and relieving poverty, but they became the new establishment. This was the new face of modern liberalism.
Then in 1984, after the Republicans had used the word "liberal" to describe positions they'd supported for decades, Ronald Reagan reframed the word to refer to his ideological opponents. It became a smear word used by those on the right. With the ascension of right-wing talk radio in the 1990's, with personalities like Rush Limbaugh and G. Gordon Liddy repeatedly using the word "liberal" to describe everything they found objectionable, it oozed into the vernacular vocabulary, and liberals started to feel attacked by the label. They started calling themselves "progressives" (as they had from 1900-1920).
Then the conservatives started sneering at all things "progressive," and that word was soon tainted with the same stench. Modern liberals needed a new word, but were having trouble coming up with one at that point. So in 2014, when Black Lives Matter popularized the term "woke," to describe what effectively was just social liberalism, the modern liberals latched onto that as their new label. And predictably, the conservatives turned that into their new swear word. Even things that had nothing to do with politics in any fashion--if conservatives didn't like a thing, they called it "woke" (much as teenagers had used the word "gay" a decade earlier).
I think this is a funny dance, because we see it repeated with the names for the people the liberals claim to champion. Bums became hobos. Hobos became "the needy." The needy became the underprivileged. The underprivileged became the homeless. And now I'm seeing hypersensitive liberals think they're making something better by ordering us all to refer to the homeless as "the unhoused," which, of course, means literally the same thing.
The polite term for people brought here from Africa used to be "negroes." Then for about a century, they went back and forth between the terms "colored" and "negro" as being preferred, each time with the non-preferred one seen as being less respectful. In the 1970s, the word "colored" fell out of favor entirely, and by the 1980s, the word "negro" followed suit. They were replaced by "black." Then with the political correctness of the 1990s, we got "African-American," and even the people so named were getting tired of the name changes by then. After that came the more inclusive "People of Color." Before the ink was even dry on POC, it got expanded to the pointlessly redundant "BIPOC" (Black, Indigenous, and People of Color), apparently just because academics didn't want to be caught using a term that was stale enough for the rest of the intelligentsia to have already adopted. It became fashion, a race to be the most avant-garde. The latest post-BIPOC trend I'm aware of is "Black," capitalized.
The funny thing about the renaming of that last group is that there's this phenomenon of erasure, where the namers publicly proclaim that the new name isn't new. It was first used hundreds of years ago, and you're just hopeless if you didn't know that it's always been the preferred usage all along. Also, War is Peace, Freedom is Slavery, and we've always been at war with Eastasia. "Gaslighting" is the term for this, but in true gaslighting fashion, it's fashionable now for gaslighters to tell their victims that the victims are misusing the word and don't know the actual definition of "gaslighting."
Fools became idiots. Idiots became imbeciles. Imbeciles became "profound" and retarded. The retarded became the mentally retarded or "slow". The mentally retarded became the developmentally disabled. Then the developmentally disabled became the neurodivergent, "on the spectrum," and the intellectually disabled. No matter what you change the name to, though, people are going to hold the unintelligent in low esteem and use whatever name they bear as an insult.
I don't think modern liberals are ever going to learn this lesson, though. They're just going to keep wearing a label until it's saturated with shame, then toss it out and put on a new one. Maybe at some point, we'll do a laundering of labels and start calling ourselves socialists again.