Privilege is interesting. I imagine it as a ladder where the top rung is very clearly defined and the every rung underneath is obscured. The position of any particular rung is based on a combination of very specific specifications (some analog and some digital) - much like a synthesizer is a combination of different settings to produce a particular sound. Synthesizers work by taking a signal and subtracting things from it to produce sound (much like a statue is carved out of a block of marble). Some examples would be races (although the actual skin color would be a a completely different analog setting), economic status, sexual identity, gender identity, family background, amongst a whole lot of other things.
Now after a non-top rung person's formula is determined we can approximate the location of that rung based on a comparison to the top rung, but we cannot with certainty and precision ascertain its position. Also, it's hard to compare where each individual rung should be in relation to each other. Furthermore, the fact that everything below the top is obscured poses a completely different problem - we don't know how many rungs there actually are on the ladder, so we can never really know how many are above or below any particular rung.
Where this all comes into play in the real world is this: different groups of people experience varying degrees of privilege depending on how similar they are to the the top rung of our privilege ladder, which I would argue is the straight, white, multigenerationally wealthy, cisgendered male with great political power and influence (there are other factors, but let's keep it slightly simple for now). Anybody who is not EXACTLY of that archetype can claim lack of a certain privilege. And everyone who is not EXACTLY of that archetype can claim some level of oppression from that archetype. Now obviously, nobody is going to lose sleep over a straight, white, multigenerationally wealthy, cisgendered male with only medium political power and influence being oppressed by the top rung, but there is definitely a hierarchy, and oppression does not only come from the very top rung.
For example, a gay, white, middle class, cisgendered male with no political power and influence can still oppress a straight, black, multigenerationally poor, cisgendered female with no political power and influence as easily as somebody could oppress him from a higher rung. Privilege is basically the removal of a specific factor that would otherwise leave you open to a oppression based on that factor. It's not one on/off switch or a sum - it's one of any number of factors, and it's absence in one category does not preclude its presence in another, just as being oppressed does not preclude a particular person from oppressing somebody else.
Where this all comes into play on the topic at hand is that nobody who is being oppressed wants to hear that they are oppressing somebody else. Nobody who lacks a certain privilege wants to hear that they have a privilege that others lack. So many white people hate it when they are called privileged - especially those who have not had it too well economically, or who have been harassed because they are not straight or cisgendered, etc. We've seen in this thread that black men don't want to accept that black women have it harder than them in terms of privilege because they are both black and women - two traditionally and institutionally oppressed groups for the same person. Whether we want to hear it or not, we still have to accept it if we are to break the damned ladder in the first place. Because the worst thing those of us who have a privilege or privileges can do is to take them for granted and ignore the pleas of those without them. It doesn't make them just as bad as those on the top rung, but it does seem a bit tone deaf and hypocritical to do so.
