Because if you're trying to convince Rory that there's nothing going on between you and his fiancee, her having the keys to your house might be a tad difficult to explain. Of course, that isn't a problem now, but the Doc's likely still feeling too guilty to worry about silly issues like keys.
I'm wondering why he still uses a key at all. He knows now all he has to do is snap his fingers. Or does he just keep that for special occassions, like when he wants to impress a girl?
Rose got hers far too easy. A companion should have to really earn it. Besides, I think the Doctor realises there is something very odd about Amy underneath it all and is holding back for that reason.
I actually agree that Rose got her too early which thinking back on it now seems strange for the Doctor's behavior. He wasn't looking for another companion only to stop the Nestene and the Autons. The fact that he seems relatively eager to travel with her after wards (he's lonely obviously) and trusting of her too to give her a key makes him seem desperate a little. Maybe he was just testing her to see how she would react. Amy doesn't need a key. She wants to travel with the Doctor and has been waiting her entire life to do so. She also takes her cues from the Doctor moreso than maybe recent companions have but isn't afraid to stand up for herself or go head strong into danger.
She doesn't need a key, she proved in the Beat Below she's quite adept at picking locks... Personally I think the Doctor is worried she might nip in and nick his telly or something...
I saw her first! Actually I didn't cos I was out when they made the announcement. My girlfriend texted me to advise she was red haired and Scottish...then said I wasn't going to be allowed to watch Dr Who anymore
From the way I remember it, Rose only got the key as a distraction method - but Martha and Jack had them too... Different Doctors, different decisions. It's all good.
Wait a minute since when did the Doctor give Jack a Tardis key and why would he when he's done his best to avoid him like the plague? Amy might be adept at picking locks but I doubt she'd be able to pick the lock on a Tardis. Also this might be obvious now but if she had been given a Tardis key the entire plot for the episode "The Lodger" tonight would be rendered nonexistent
Jack let himself into the TARDIS in "Bad Wolf"; that's how he figured out where Rose went. In "Vampires of Venice" the Doctor tells Amy and Rory to go wait for him in the TARDIS, so maybe she does have a key and they just didn't make a big production out of him giving it to her? Also, at the end of that episode she just opens the door; the Doctor never unlocked it for her. Maybe the TARDIS just lets in whoever she wants, and the key isn't really needed at all. She did lock the Doctor out in "The 11th Hour".
And he still has the key centuries later in "The Sound of Drums," since the Doctor uses it to make Jack a perception filter. Pretty impressive!