But if you want to select a date relevant to the era, it would be 1356 - the date of the Diet of Nuremburg that resulted in the Golden Bull, which established the constitutional structure of the Holy Roman Empire until its demise, and was the last major change to the European system until the Thirty Years' War and the Peace of Westphalia.
Of course, if you don't want to be Euro-centric, you might pick other dates. Timur's armies sacked Delhi on December 17th, 1398; the Mughal Empire was established by Babur in 1526 and eventually established the last stable empire on the subcontinent before British rule. The Yuan Dynasty of Mongols in China was overthrown by the Han Chinese Ming Dynasty in 1368, which proceeded to seek further contact with the outside world, resulting in Admiral Zheng He's voyages before the Confucian elites prevailed on the Emperor to stop wasting money showing off boats to barbarians. The Aztecs established hegemony over the native tribes of what we now call Mexico in the 1420s under Tlacaelel. The Inca began their imperial expansion in ~1438 under Sapa Inca Pachacuti. The Great Khmer Empire in Southeast Asia reached its watershed around 1327. Sarai, the capital of the Golden Horde and one of the largest cities of the medieval world, was destroyed in 1556. The Lithuanian king Jogaila converted to Christianity in 1385. The Battle of the Ice in 1242 effectively ended the Northern Crusaders' battle against the Russian Orthodox states. And so on and so forth.
You can pick whatever date you'd like and make a case for it. Personally I'd pick 1369 or so: the Ming Dynasty is just beginning to rule China, and will do so for most of the scenario. The Golden Bull has just been issued, establishing the Holy Roman Empire as it is represented in the game; Lithuania has just converted to Catholicism the prior year and the Northern Crusades are ended, marking the end of the crusader period in the north; Timur has just been proclaimed sovereign in Samarkand; the Ottomans have just arrived in Europe, having won the Battle of Adrianople in 1365. It's a point where many of the key areas of focus have just entered crucial periods that Europa Universalis seeks to enact.