"Update profile" seems more in the style of Computron 5000 to me, but yeah, the move to "update my profile" was an attempt to make it friendly and now it has that connotation. But I think it works mostly by making the idea of a dialog (in the sense of conversation) more explicit, not so much about ownership. Like you also started seeing prompts like "Would you like to...?" or "Are you sure you want to delete?" rather than "Confirm deletion."

Dialog works because my perception of my interaction with software is that I'm in a dialog with it. When the interface mirrors that idea back to me, there's a fit or congruence and it feels comfortable, in the same way that an ergonomic handle is something like the inverse of the shape of your hand while holding it - I say "my profile" so the interface should say the linguistic inverse: "your profile". IMO, it has nothing to do with humanness or friendliness or emotion -- which is nice if it's contextually appropriate, but it can be irritating -- and more to do with cognitive ergonomics.

The idea of making computers more friendly, like a human, to make them easier to use is kind of a dumb idea when you think about it, like saying screwdriver handles should be shaped like human hands because we find it comfortable to hold someone's hand. Anyway, human-to-human interaction is often very awkward and definitely not intuitive, why should it be the ideal for human-computer interaction?