"Let me not to the marriage of true minds. Admit impediments. Love is not love. Which alters when it alteration finds, Or bends with the remover to remove: O no! it is an ever-fixed mark, That looks on tempests and is never shaken; It is the star to ever wandering bark, Whose worth's unknown, although his height be taken. Love's not Time's fool, though rosy lips and cheeks...Within his bending sickle's compass come: Love alters not with his brief hours and weeks, But bears it out even to the edge of doom. If this be error and upon me proved, I never writ, nor no man ever loved." -William Shakespeare
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.