They can claim whatever they want and they can call their country as they wish.
This is their sovereign country, this is their land and they have every right to do whatever they want.
They do exactly that! They claim Alexander the Great and the ancient-Macedonians from antiquity to be their hereditary ancestors and humiliate Greeks in the process. They also claim that Greece is occupying Macedonian territory.
FYRoM does exactly what you state above with no regard to the consequences of those actions. That kind of behaviour is considered confrontationalist, not conducive to good neighbourly relations...a pre-requisite for joining the EU and NATO.
All outstanding issues with neighbouring countries FYRoM would need to resolve before gaining entry into the Western Worlds most prestigious economic and security structures.
FYRoM has problems with all of it's neighbours except Turkey!
In FYRoM live Slavic-speaking Serbo-Bulgar peoples wanting to self-determine as Macedonians...Ethnically, Racially, Nationally, Culturally and Linguistically, but Macedonians have always been Greek-speaking Hellenic-peoples...Northern-Greeks, from the ancient-Kingdom of Macedon which is located in Northern-Greece. Macedonian history belongs to no-one but the Greeks and their legacy belongs to Greek heritage.
Had FYRoM's ex-Yugoslavs introduced themselves to the world as Slavs but Macedonians too, Greeks would have sat down with them to discuss exactly, their Macedonian credentials, but they didn't do that...instead, they told the world that Alexander the Great was the first Czar of the Slavs and ancient-Macedonians were proto-Slavic peoples. They commissioned a Spanish Haematologist to doctor the results of a genetic study which placed the Greek sample in the African cluster. They used genetic propaganda to show Greeks had African origins whilst they were indigenous and native to the Haemus-peninsula. Their Maps show Macedonian territory in Greece as Occupied territory
and stage events depicting the funeral of Greece. They have even erased the White Cross on the Greek flag and replaced it with a Swastika. Greeks have been sensitized to FYRoM's propaganda antics so every word they say there, and every step they take there, is scrutinized to the nth, degree.
Macedonia for the Greeks constitues a large part of their geography and an even larger part of their national-history and cultural-heritage...we aim to keep all Macedonian Identity factors Greek-Hellenic, because we owe it to our ancestral forebears to do that.
FYRoM is not Macedonia and the peoples there are Slavic. On the other hand, Macedonians have always been Greek-speaking Hellenic-peoples from since the days of King Karanus 778-808 BC. FYRoM's self-determination right has to be weighed-up against the cultural-historical rights of Greeks wanting to continue to self determine in the tradition of their regional historical ancestors.