Why would a country call itself that?
Because of its southern neighbour, Greece. Macedonia is the name of a historical greek province, and the greeks apparently believe that a country will call itself “Macedonia” so that it can claim the greek province with the same name. You can read more about this on Wikipedia.
Sort of like Canada not recognizing USA because, hey, a country named United States of America must want to conquer the whole America…it’s right there in the name, after all. 
BTW the USA has been recognizing them as plain old Republic of Macedonia for years now.
But the various countries, including Macedonian itself, have generally decided to do the polite, PC thing and when in international activities involving both Macedonia and Greece, humor the latter by going for the FYROM designation (sort of like how Taiwan/ROC is referred to as “Chinese Taipei”).
Dan_ch: Or the Mexican and USA states of California and Baja California demanding of each other they change their names because there can only be one California and it’s us.
They do it as a diplomatic courtesy to the insistence that they do it of the Former Turkish Nation of Greece. 
By the way, I feel that the proper term for an inhabitant of FYROM is “FYROManiac.”
In the business world, one would say: The name Macedonia is trademarked.
That’s Former Ottoman Nation of Greece – “FONG” for short.
There really is an irredentist movementwanting to get New Mexico back for Mexico. Along with the rest of the US Southwest, of course.
On a side note: Why is it pronounced masedonia in English instead of makedonia (which is the local pronunciation)?
Perhaps because it came into English via Greek, not Macedonian. In Greek, the second consonant is not the IPA /k/ but the IPA /c/.
It’s worse than that - the Greeks don’t believe that the Macedonians are actual Macedonians, but rather the Slavic usurpers of the *real *Macedonians, who were actually Greeks.
Mind you, right about now FYR Macedonia could probably buy the Greek province off them at a good price.
Are you sure? It is written with a K in Greek - is that letter really pronounced with a “ch”?
My guess would have been that it was pronounced Makedonia in classical Latin, became Machedonia in later Latin and Italian, then Masedwan in French, and came into English with the /s/ sound from French.