Southern (USA) accent is made fun of and not accepted even in some parts of America like California, that's what southerners say
You can see how southern accent is mocked even here
So it's not just people of color who go thru this
Not just the southern accent. The hillbilly/Appalachian accent is made fun of a lot. And as a Midwestern, I'm well aware that the Midwestern accent, especially Yooper, is treated as a punchline. People from the Midwest are generally perceived as simple, unwitting, slow, and not worldly. Little do they know Midwest nice is sugarcoated burn - ain't that special.
https://www.youtube.com/watch?v=1JDHSoTAwgg
The entire USA is filled with regional biases and rivalries. So yes, many white people do get made fun of a lot. Even wealthy white folks from the Hamptons get made fun of for their mannerisms and accents.
But at least for most groups, they have a sense of belonging and majority status somewhere in the country. It hits very differently when you don't just sound different but look different too.