What do you call a person who behaves in an overly familiar way, taking too many liberties, saying things that only someone with a closer relationship with you would normally say?
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oMiKey is close, but I believe this question requires a term with a slightly more specific implication. I would say this person is 'forward', from Webster:
Lacking modesty or reserve. Brash, poorly disciplined children are often distressingly forward.
They are being over-familiar - Too familiar. Inappropriately or unduly intimate or informal (OED).
Impertinent - (Of persons, their actions, etc.) Meddling with what is beyond one's province; intrusive, presumptuous; behaving without proper respect or deference to superiors or strangers; insolent or saucy in speech or behaviour (OED).
OR (just plain),
Rude - Unmannerly, uncivil, impolite; offensively or deliberately discourteous.(OED)
The word you want is presumptuous: (link and defintion from Merriam-Webster)
overstepping due bounds (as of propriety or courtesy) : taking liberties
The word shares a root with presume, suggesting a person who presumes to know things better or to speak to people as they though were more familiar. As an example:
It was very presumptuous of Bob to ask Alice about the details of her vacation plans.
Presumptuous - (of a person or their behavior) failing to observe the limits of what is permitted or appropriate. "I hope I won't be considered presumptuous if I offer some advice"
pert - boldly forward in speech or behavior; impertinent;
impertinent - intrusive or presumptuous,; brash, inappropriate
Depending on the context, smitten or love-struck could fit.
I would say the person is 'nonchalant'.
(of a person or manner) feeling or appearing casually calm and relaxed; not displaying anxiety, interest, or enthusiasm.