I got the impression

Hmm but where did you get that impression from exactly? Reading theory, or cultural osmosis? The right loves to use the term in a very vague and poorly understood way that basically comes down to whatever trends they dislike, and if you're absorbing the meaning from cultural osmosis then you're probably getting some of that influence in there.

The reality is that while intersectionality does concern itself with problems that are unique to specific intersections of identity, such a focus allows for a broader level of inclusion, and avoids generalizing over people's experiences. Intersectionality originates as a response to a tendency in early feminism to focus on the perspectives, experiences, and priorities of middle-class white women, and treating those things as universal. Non-intersectional feminism is still concerned with problems that concern people at a specific intersection of gender, race, and class, but it is only concerned with one specific intersection (i.e. women, white, and middle-class). Intersectionality is not about establishing a hierarchy of victimhood or saying that the more axes of oppression you're on the more valid your opinion is, rather, it simply allows for the recognition that the forms of oppression that people encounter are often varied and complex, and are not simply the sum of the parts. A black woman may encounter specific forms of discrimination that are not experienced by either a white woman or a black man. And there's stuff that a white woman might experience that neither a black woman nor a white man would.

Side note: there was a :bait: meme that went around back in the day (2010-ish maybe?) that assigned point values to various traits and you were supposed to add them up and the total would tell you how privileged you are. It was widely mocked (you get -500 points of privilege if you're trans, but instead get +10 if you're trans (passable) lmao) and it was often associated with the term intersectionality, but if you read anything I just wrote, then you should recognize that this is totally inconsistent with the concept of intersectionality. In ways that critics often pointed out! I remember people at this time loved to use the example that if you were a Jewish person living in a predominately Muslim country that hated Jews, you'd score as more privileged than a Muslim living in the same country. Almost as if the traits of religion and nationality... intersect in ways that are more complicated than just adding them together! It's funny but chuds will reinvent intersectionality while trying to dunk on what they think intersectionality is because they have zero theory and they're often dunking on libs who also have zero theory. Read theory!

So why is it important to acknowledge the unique differences that emerge from different intersections of traits? Well, first off because it's true. But also because recognizing these experiences means recognizing the value that different people's perspectives can add to a conversation. You can get a more complete picture of the world and recognize more ways in which discrimination can manifest by including more perspectives. By listening to the concerns of specific communities and people with specific intersections of identity, a movement can define goals in a way that has a broader appeal, and avoids making people feel alienated as they might be if your group was only concerned with its own largest demographic group. It is not necessary, or possible, to be everything to everyone and it's good to keep practical concerns in mind, but making an effort to hear people out can help to be more inclusive and expansive.