I think that a lot of the reason Jarvis has become so human is because Tony treats him like he’s human. Tony talks to Jarvis in a very colloquial way. He says “you up?” when he knows damn well that Jarvis is operational. He says “throw a little hot-rod red in there” instead of “paint components x, y, and z with red paint #20.” Tony treats all his machinery like that—Dummy and You, especially—and Jarvis is no exception.
Jarvis has become much more human since Iron Man 1. He actually displayed emotions in Iron Man 3—specifically when he feared for Tony’s life, his voice sounded terribly frightened, and in instances like the second gif where he said “I need to sleep” and not “My battery is depleted.” Jarvis has grown and changed, as any self-aware creature does. He has become human because he is treated as such.