Reading debates like this makes me sad. I realize that just like everything else, feminism also is a tool that different people can use for different purposes. Some people can use it to support empathy towards other human beings. Some people use it to deny empathy towards other human beings. Somehow the latter seem more prominent on the internet.

There is something in the impersonal online communication that emphasises sociopathic traits in people. When in real life you see a suffering man, you usually don't see feminists running to him screaming "actually, I have it much worse!". But when you see a man writing about his suffering online, this is what often happens. Maybe it's because behaving like an asshole in real life is likely to bring you a punch in the face or at least a loss of status, while doing the same on internet generates page views and income, if you do it properly.

Like polymathwannabe wrote, there is a meaningful definition of "privilege" (although I would prefer the word "advantage"), which is: "Maybe you are just a lucky person who doesn't have this problem, so you don't think about this problem, and you may even find it unbelievable, but for some other people this is an important factor in their lives." This is the core that makes sense.

The part of the online discource that doesn't make sense is the frequent claim that this "privilege" is unidirectional: "If you separate people into two groups X and Y, exactly one of them will have problems that the other side doesn't see." There is an illogical jump from "group X has on average more problems than group Y" to "problems of the group X are a strict superset of problems of the group Y". This implicit assumption manifests in a debate when people from group Y are told to "check their privilege", but is assumed that people from group X have no provilege to check. (Unless we consider some other dimension, so e.g. people from X1 may have privilege over people from X2, but still never a privilege over Y1.)

Like Sarunas described, the problem is replacing the reality with a simplified multilinear model. First problem, the model depends on which variables are included and which are excluded, which is somewhat an arbitrary decision, so of course people are prone to include the variables which "prove" their oppression, and exclude the variables which "disprove" it. (For example, there is usually a strong emphasis on "male privilege", but very little emphasis on "rich privilege". Which is coincidentally exactly the kind of bias you would expect if the theory is popular among female Harvard students.) Second problem, the multilinear assumption itself may be wrong, even if we include all the relevant variables. Third problem, even if the multilinear model would descibe the averages of groups accurately, individuals are not the averages of their groups. It doesn't make much sense to assume that a random black homeless man has a lot of power and money just because a different individual from the "black male" group happens to be a US president.