Furthermore, I assume ancient Greek is not mutually intelligible with modern Greek, similarly to modern English and ancient English (correct me if I'm wrong).
Why is it still considered the same language where Italian, Spanish, Portuguese, etc are not considered to be the "Latin" language?
French, Italian, Spanish etc. aren't called the Latin Languages because they're already called the Romance languages. They're all variations of Latin filtered through centuries of changing pronunciations by various Germanic or "Celtic" tribes.
Following the fall of Rome, large areas of Italy were conquered by the Lombards, the Byzantines and later the Carolingians, then Ottonians and Normans. In Spain, the Visigoths established a kingdom based around Toledo. In Gaul, much of the country is taken by the Franks, with other areas Flemish, Romano-British and later Norman. Everybody speaks their own variations of Latin, mixed with their own languages, loan words, provincial dialects, and so the various languages gradually drift from Latin. As an example of how quickly this can happen, following the Norman Conquest of England in 1066, the French spoken there becomes a unique mixture of Norman, Picardy, Frankish and Breton dialects with some borrowed Langue d'Oc grammar and a healthy dose of Old English thrown into the mix that within a generation is completely different from the French being spoken even in Normandy.
Out of interest, until roughly the Albigensian Crusade, much of the south of France in the Occitan region spoke Langue d'Oc, a variant of French much closer to Catalan than the Germanic Langue d'Or spoken in Frankish France, and linguistically much closer to original Latin.