I think this has to do with the difference between mass and count nouns, but I am not sure.
So English makes a distinction between things and stuff. Things come in individual chunks. So, for example, you can count them. I can have three drinks, or five friends, or three left shoes. Stuff is some sort of substance or, we might say, it is continuous. It does not come in individual chunks, unless you measure those out. So for instance, I cannot breathe "three airs." I could maybe breathe three liters of air, measuring it out. Sometimes for the really common stuff, it becomes things because it is always measured in just one way. So for example water and coffee and sugar and cream are all stuffs, but in a restaurant, a waiter might say "table three needs two waters and three coffees, two black, one with two sugars and three creams." He is able to play with this aspect of grammar, because in that context he means two glasses of water, three cups of coffee, one of which needs two packets of sugar and three containers of cream.
Similarly if you were to say "She has two salts" it is not grammatical at first, and in my attempts to make it grammatical in my head I start to hope that we are talking about a chemist who has two different kinds of salt, or so, maybe she is mixing sodium chloride and potassium iodide for example.
Now, throats are things. They are things even when they are sore.
Throats can be sore for a lot of reasons. Maybe you were screaming all night last night at a concert and now you have a sore throat. Maybe your throat was recently operated on in a surgery to remove your tonsils and it is still sore.
Strep is a particular bacterium that causes a particular disease, known as strep throat. Diseases like strep throat and cancer are usually thought of as stuff, not as things. So much like the "She has two salts" example above where I am hoping she is a chemist, if you tell me "He has three cancers", that sounds very unfortunate, because I have to imagine that you are talking about different kinds of cancer, "yes, brain cancer, pancreatic cancer, and prostate cancer." That's a very sick man indeed! The words we use to measure these conditions are somewhat different; I have to speak of cases of cancer or so, because it is such an unusual substance.