Why does the Republic of Macedonia have its name when the historical Macedonia is in Greece?

There is a 19th century answer to the question. In the eastern Mediterranean, place names changed organically over the centuries. These names were often corruptions of Greek or Roman names, or the Arabic name, or whatever. This is how names like the southern Greek Peninsula becoming known as Morea, Iconium becoming Konya, or Syria becoming Sham came to be. It happened all over the place in Byzantine/Arab/Ottoman lands.

Anyways fast forward to the late 18th/ early 19th centuries. Western Europeans began coming east quite a bit and "discovering" Ottoman lands. Except, they knew all these places by their Biblical or Classical era names, because unsurprisingly they had most of their knowledge of the place from the Bible or learning Latin or Greek as an educated individual. Those names were often quite different from the local names which had been organically developed over centuries. The Europeans insisted on using certain names for places from their classical works, and Ottoman officials began following suit.

You can see the shifting Ottoman usage of names in the Ottoman province reorginization in 1864. They wanted to normalize administration following the French model. Sham became Suriye, which was the classical name for the region. Just one example of the Ottomans starting to normalize their terms for places with what Europeans called them. This fits the general Ottoman trend of adopting and internalizing words and concepts from Europe, such as them coining their own words for "independence" or "constitution," or any other number of things you can read about in any book on the Tanzimat.

This was a longstanding trend. Europeans started calling the southern Balkans "Macedonia" because, well, that was the classical name for the region. The ethnic situation there was EXTREMELY murky in the late 19th century, and anyone who tells you it was not probably has a nationalist agenda. Muslims, Jews, Albanians and Christians speaking Bulgarian, Greek, or Serbian all lived in the area. In 1878 Ottoman Europe shrank drastically following a war with Russia. The concept of "Macedonia" then expanded from "classical" Macedonia to basically the remaining Ottoman territory in Europe. Look up the borders of the empire from 1878-1911 to see the area I'm talking about.

This region became known as "Ottoman Macedonia." The Europeans called it that, and it was pretty common for the Ottomans to call it that as well. Again, this was after a century of Europeans insisting on calling places by their classical names, and slow Ottoman internalization of that. The Ottomans also referred to the area as "The Three Provinces" (Salnica, Monastir, and Kosovo). The European powers were ultra interested because of the "Eastern Question" generally, which was the fear of Russia getting the lion's share of a collapsed Ottoman state, and the "Macedonian Question" particularly, which was a concern over the ethnic strife in the region. Greek and Bulgarian nationalists were fighting insurgencies in the region while Ottoman troops ran around trying to put them down. Massacres were committed on all sides, and it was a very dicey situation in general.

Eventually the strife boiled over in the 1912-1913 Balkan Wars. The Balkan countries attacked the Ottoman Empire and met with great success. Such great success, in fact, that they immediately fell on each other to divide the territorial spoils. Territory not too different from today's borders was established. The populations, however, were still very ethnically mixed. During the war and after there was a fair amount of ethnic cleansing to make sure the populations matched the new borders.

Anyways, the Serbians ended up with a chunk of Macedonia that was pretty far west. Its population spoke a language closer to Bulgarian than Serbian. In fact, Russia had tried to annex the region into Bulgaria in 1878 before being overruled at the Congress of Berlin. The Serbians had to come up with a way to stress that this chunk of land and its population was separate from Bulgaria. As it happens, they basically just kept the name from the Ottoman era- they just called the region of Ottoman Macedonia that they had annexed Macedonia. They then encouraged the creation of a separate (Slavic) Macedonian identity. Nobody much noticed until Yugoslavia broke up and Greece used its power in the EU to raise a ruckus.

TL;DR Ottomans started naming stuff after classical places. Macedonia was recovered as a name for the southern Balkans. It became a larger term for all of Ottoman Europe after 1878. In 1913 Serbia kept the name for their chunk of conquered Ottoman Macedonia. Serbia had to differentiate this area from the Bulgarians, and so fostered a Slavic Macedonian Identity.