The answer is 1991-1993, when the republic of Macedonia declared it's independence.
For an in-depth understanding of the issue, one would have to examine the origins and development of ethnic and national identities in the Balkans. Many of the political developments fall after the 1998 cut-off date, so I will concentrate on providing some background and the outlining the political developments in the early 90s.
What this question is asking can be rephrased as when each side started claiming ownership of a Macedonian identity, understood as a connection with an ancient Macedonian kingdom. Since it's a question about the construction of a narrative around which an ethnic identity is built, whether ancient Macedonians identified as Greeks or not is besides the point.
This is important since most of the discussion in both countries usually devolves into that.
The question therefore is: when did Macedonians started identifying as such, and when did Greeks started identifying a Macedonian identity as exclusively Greek.
From the Macedonian side the answer is relatively easy. From it's origin in the 19th century, Macedonian nationalism claimed that identity and a connection to a Macedonian heritage, around which the Macedonian identity was constructed.
During the rise of balkan nationalisms in the 19th century, the establishment of the nation states in the balkans, Macedonians were one of the few ethnic groups that did not manage to form an independent nation state. Macedonia was first established as a socialist republic of the Yugoslav Federation. It is important to note that Macedonia inlcuded a significant ethnic albanian minority. Additionally, the region was considered for the Albanians part of "Great Albania", for the Bulgarians a province of Bulgaria, and for the Greeks part of the so-called "Big idea" of a Greece reaching all the way to Asia minor and the Caucasus. Macedonian nationalism is relatively more constrained, generally claiming only what is now northen greece, western bulgaria and Kosovo. Welcome to balkan nationalism.
From Greek sources during the Balkan wars in the early 20th century, we can see they considered Macedonians as a distinct ethnic group separate from the greek identity. One such example can be found the memoirs of Pavlos Melas, a greek army officer that participated in paramilitary action in the balkans at the time.
So one can conclude that in the early 20th century Greeks did not care much about "re"-claiming the macedonian identity, name or heritage from what was they considered macedonian population at the time.
During the Greek Civil War (1945-1949) the issue comes up again when part of the Macedonian minority in northern Greece participates (willingly or not) in the conflict on the side of the Democratic Army, which was affiliated with the greek communist party and was provided some support from Yugoslavia. While the other side refers to the macedonian population as "traitors" to the national cause, they do not appear bothered by the use of the term macedonian to describe that ethnic group. A good source on that is the "History of the Greek Civil War 1946-1949" by Giorgos Margaritis.
During the post-war period, Macedonia became a socialist republic of the Yugoslav federation. While the macedonian minority in northen Greece did suffer during that period, the name or the heritage were not political issues in any major way. On the Greek side the macedonian heritage was considered part of the big narrative of the greek national identity, and nobody bothered to challege that. Macedonia was also a socialist republic of yugoslavia so that was that.
When Macedonia declared its idependence during the collapse of yugoslavia, it became a political issue in greece since it was percieved as a direct claim and challenge of the ownership of that identity. Or at least it presented itself as an issue that could be used to stoke the nationalist sentiment and provide a political advantage domestically, as what happened in the following years clearly shows.
I will go into a bit more depth on the political developments from the Greek side, since they were the more consequential.
The greek goverement at the time was headed by K. Mitsotakis and his right wing party "New Democracy" with a very slim majority. He adopted a moderate and pragmatic line towards the issue favoring the stability of the region over domestic political concerns. That ended up costing him dearly as huge demonstrations started taking place in 1992 calling for a more hard line. A. Samaras, a high ranking member of the governing party seized the opportunity and, declaring any compromise as betrayal, broke away from the party forming his own short-lived political party, depriving the goverment of a working majority. The goverment fell leading to snap elections in 1993. The political climate was such that any attempt at hinting at a compromise was considered nothing sort of treasonous, and made adopting any softer position political suicide. This was exploited by the opposition socialist party, with its leader A. Papandreou adopting an even more confrontational platform and winning the elections. The Macedonian side would react in a similar way, ramping up similar sentiments, although pragmatic compromise was considered more viable.
The Greek goverment would block any attempt by the republic of macedonia to join any international body in any possible way including the UN, EU, NATO and even FIFA. In the midst of a nationalistic fervor that continues to this day, any political position that hints at a compromise remained and still remains incredibly precarious. By the end of the 90s Greece was still managing, under threat of veto, to keep the republic of Macedonia from joining the EU and NATO. Under an interim agreement, Greece recognized the name FYROM (Former Yugoslav Republic Of Macedonia), but refused and avoided any use of the term Republic of Macedonia, as well as references to a Macedonian language, ethinicity or nationality. Macedonia could not use the the name "Republic of Macedonia" when Greece was involved i.e. everywhere except the EU and NATO, where Greece held veto power, and border crossing visa stamps for Greek passports (since greek authorities would not recognize a visa stamp that included the term "Republic of Macedonia"). This was the status quo in the late 90s, and largely remains unchanged since then up until a few months ago but that's off topic.
To conclude, while the issue of ownership of a macedonian heritage was a central element throughout the 19th and 20th century for both Macedonian and Greek nationalism, it did not evolve to an active confrontation until the early 90s when both sides had nation states that desperately needed to protect their respective narratives that were at the core of each national identity.
EDIT: Added some clarification on the situation by the end of the 90s