This is a problem of nationalism. Not just Macedonian or Greek nationalism, but in general.
So the reasons for Greek nationalists is really exactly the same as for FYRO-Macedonians:
Macedonian President 'Won't Sign' Name Deal With Greece:
Macedonian President Gjorge Ivanov has said he won't sign a deal with Greece on the former Yugoslav republic's name, saying it violates the country's constitution.
Ivanov made the announcement on June 13, a day after the Macedonian and Greek prime ministers announced they had reached an agreement to name the country Republic of North Macedonia, or Severna Makedonija in Macedonian.
Under the accord, its language will be Macedonian and its people known as Macedonians or citizens of the Republic of North Macedonia.
"My position is final and I will not yield to any pressure, blackmail, or threats. I will not support or sign such a damaging agreement," Ivanov told a news conference.
Later in the day, about 1,000 people gathered in front of the parliament building in Skopje, calling for Prime Minister Zoran Zaev's resignation.
The demonstrators were holding Macedonian flags and chanting slogans such as "Zaev is a traitor."
The region of Macedonia is not identical with any state today:
Macedonia is a geographical and historical region of the Balkan Peninsula in Southeast Europe. Its boundaries have changed considerably over time; however, it came to be defined as the modern geographical region by the mid 19th century. Nowadays the region is considered to include parts of six Balkan countries: Greece, the Republic of Macedonia, Bulgaria, Albania, Serbia and Kosovo.[a] It covers approximately 67,000 square kilometres (25,869 sq mi) and has a population of 4.76 million.
The historically untenable appeal is of course that "Macedonians" are those "children" of the great Alexander we all know.
But laying claim to that title is as worryingly warringly ambitious as said Alexander himself. And that's because of:
Irredentism is any political or popular movement that seeks to claim/reclaim and occupy a land that the movement's members consider to be a "lost" (or "unredeemed") territory from their nation's past.
The geographical qualifier would be a compromise that implies different Macedonias exist, but doesn't quell the possibility for irredentism. After all, if there is a North, where is the South; and why are they not together? After all there is the hidden monster called United Macedonia:
The United Macedonia concept is still found among official sources in the Republic, and taught in schools through school textbooks and through other governmental publications.
Nationalism as an ideology is unable to resolve this issue at the fundamental level. From both sides. From all sides.
The irony on this is of course that now Greeks consider Alexander and ancient Macedonians as part of their heritage, whereas before Philip the Hellenes considered the Macedons as half-barbaric cousins-at-best to the North. Modern day Macedonians are not even cousins to Greeks as that area is settled by slavic people not speaking a dialect of Greek. People engagung in Greek nationalism therefore want the name, maybe the land but certainly not the people living there now.
Incompatible world views and irreconcilable positions, that's patriotic nationalism.
Nationalism is another antonym to rationalism.
The reason for this is that the basic characteristic of the name issue between Macedonia and Greece is its symbolism, ‘signage’; this is an issue involving the use and control of symbols that signify identity and relate to the word ‘Macedonia and Macedonian’. For this, both countries have pleaded, on different grounds, their ‘ownership’, control or participation in its meaning. In this regard, this dispute is very postmodern, a dispute about symbols, ironical one, constructing simulacra of history interpretation.
You could not understand, for example, the Greek coinage about ‘stealing history’ that allegedly ‘the Skopians” do to them, or the Macedonian paranoia that Greeks actually do not like that Macedonia and Macedonians exist at all.
If one puts aside the baroque‐like Balkan ornaments and arrogance of the Balkan small imperialisms such as the Greek one, there are theses which finally become accessible when reducing ‘the Greek position’ in the issue seen from this aspect.
For example, it is possible to reach ‘the Greek fear’ that if the Greeks recognize or there is wider affirmation of a country with the name of Macedonia and of people with same such name (regardless of the fact how much the word ‘Macedonia’ would be used inside Greece for Greek Macedonians and in the northern Greek province), then the notion of ‘MACEDONIA AND MACEDONIAN’ in international relations will depart together with us, the Macedonians from the Republic of Macedonia.
Greece cannot use internationally, even if it wants so, the term Macedonia and Macedonians for its own citizens, because it has a hysterically ethno‐homogeneous orientation towards a single Greek nation and ethnos. Hence, Greece uses only internally and geographically the term Macedonia or Greek Macedonians.
This means that de facto we will be the only Macedonians and our state the only Macedonia internationally. This is a nightmare for the Greeks, a nightmare they try to hinder by all means.
The latest variation of such “impeding” reasoning the Greek position is built on the subject that Macedonians, if bearing such identity, will provoke wide‐ranging chaos in the various so‐called Macedonian identities in the Region. In line with this position there are several identities such as Greek Macedonians (Macedonian), Bulgarian Macedonians, Albanian Macedonians and Slav Macedonians in the Republic of Macedonia. This position evidently involves intentional error in confusing secondary identity of population drawn from the territory of part of the countries where they live (in that sense even the Prime minister of Greece might be Macedonian, but with Greek national identity) and primary ethnic, national identity of the Macedonians in the Republic of Macedonia.
However, the basic question for the Greek side, important in understanding the issue, is: why to divide the meanings at all? Why is the common use of the symbols Macedonia and Macedonians so important for them to preserve it as monopoly? Why would they risk to be blamed internationally and open such long low‐level crisis in the Region?
This question cannot be understood without knowing the history of ‘the Greek success’ to become part of the EU only on basis of ‘controlling’ the license of the ancient Greek democracy and culture, and not on basis of fulfilling the economic criteria for EU membership (in those times when Greece joined the EU, the former SFRY had much better economic and financial performances for joining the EU then those of Greece). Regardless this fact, Greece managed to become prominent and ‘profitable’ part of the EU, just acting on basis of ‘the obligation’ of the European nations to accept a new member that controls the territory of the ancient Greek and Latin foundations of the European civilization. This experience has been implanted in the collective perception of the Greek culture vis‐à‐ vis the external world.
Now, when ‘threat’ appears to share part of that culture with a country from the neighbourhood, the instincts of defence, based
on the experience of high profitability from the culture licenses, have come to surface in a hysterical deluge. The Greeks are especially irritated that this might happen in relation to a small country, which according to the traditional Balkan principles must be inferior and must ‘obey’ bigger regional powers.
Such farce of local ‘imperial cultures’ and their affectation constitutes the kitsch side of this dispute even today.
Danailov Ljubomir Frčkoski: "The character of the name dispute between Macedonia and Greece", Skopje: Progres Institute for Social Democracy, Skopje; Friedrich-Ebert-Stiftung, Büro Mazedonien. https://nbn-resolving.org/urn:nbn:de:0168-ssoar-63593 / Emphasis added for impatient SE readers.
Now consider how long this ulcerating boil of nationalist hatred is waiting to explode:
Nadine Lange-Akhund: "The Macedonian Question, 1893-1908, from Western Sources", Columbia University Press: New York, 1998.
Evangelos Kofos: "The Macedonian Question: The politics of mutation" Balkan Studies; Thessalonike Vol 27, No. 1, (Jan 1, 1986): 157-172. (Considering this decades old problem on the way out in 1966!)
Victor Roudometof: "Collective Memory, National Identity, and Ethnic Conflict: Greece, Bulgaria, and the Macedonian Question", Greenwood, 2002.
Kyril Drezov: "Macedonian identity: an overview of the major claims" in: The New Macedonian Question pp 47-59 In: James Pettifer (eds): "The New Macedonian Question", St Antony’s Series. Palgrave Macmillan, London, 1999. (DOI)