You got the timeline wrong.

The break-up of Yugoslavia was a long, arguably still on-going, process, the final phase of which happened peacefully. Serbia and Montenegro, that made the post-1992 Yugoslavia, agreed in 2003 to change the name of the country to Serbia and Montenegro, pending the Montenegrin independence referendum scheduled for 2006.

Considering the possibility of another country name depreciation in three years, they agreed to keep the yu domain.

Fun fact, had the Montenegrin referendum gone the other way, the plan was to use .cs as the national domain, which used to be owned by another ex-country, Czechoslovakia.

I assume you're referring to Tito? He died in 1980. None of the constituent countries tried to leave Yugoslavia until 1991, right? That's following, technically, but there's a lot of history in that decade. From my very vibes based knowledge of the area, Tito is the only dude who could have held it together though.

.yu was purchasable long after the country ceased to exist, until 2008 to be exact.

Technically speaking, "Yugoslavia" continued to exist until 2003, when the name finally got deprecated in favour of "Serbia & Montenegro" as one country (also including the territory of Kosovo), which itself only lasted 3 years before Montenegro declared independence (and Kosovo did the same 2 years after).

So however you spin it, the domain outlived the country by at least 5 years, arguably 15(ish), 9 of which were post-war(s).

The organization that ran the nameservers for .yu still exists today. Even in the case where there was no one fit to run them, all the records could be transferred to ICANN or someone else to run the server.