The first meeting, yea, Moffat probably believed Tennant would be having at least 1 adventure with her.
The "Just Who are..." makes perfect sense. She knows everything about him, but, he knows nothing about who she actually is, or why she knows so much about him. Imagine a "Hi...Bye" kind of relationship you might have with someone you see at a store once a month, that you've never had an in depth conversation with that knows your full name, your kids names, your address, etc. Wouldn't you wonder exactly who this person actually is, and why they know so much about you? You might think they were some kind of crazy stalker.
One Day I hope to be the Man my Cat thinks I am
Where are we going? And why are we in this Handbasket?