Or, which modern languages approximate the similarity between the Latin spoken in Rome and the Koine Greek that was also used as the lingua franca of the day? I’m thinking of the time period around the beginning of the Common Era.
That’s a tough question to answer. The alphabets are different, obviously. The grammatical system is in essence the same, but there’s an extra mood in Greek. The vocabulary is very different - few words sound similar, like between French and Spanish, for example. For me, at least, knowing Latin grammar was a big help in learning Greek, but it was in no way, shape or form easy.
But here’s the big point - any Roman with aspirations to politics, literature, war, etc. would have learned Greek at a very early age as a prerequisite of his social class. Learning it so early makes it a good deal easier.
A child can learn any language fluently if exposed early enough. Any boy (& possibley a few girls) who’s parents were able to educate him would have learned Greek.
It’s also important to remember that well-to-do families (when they could afford it) would purchase a Greek slave specifically to educate their children. The slave who held this esteemed post was called the pedagogue.