If you're going to pose as a commoner, you should do it properly.
That one little sentence would seem to indicate he has figured it out. Tywin, it should be noted, is always shown as having a formidable intellect coupled with cunning and a chilling lack of empathy for others. That does not mean that he disregards their feelings; he knows that others feel, and uses their feelings to manipulate and, well, rule them. He is very, very, very good at what he does. Frankly, given the doofuses who have taken the throne, he probably would have been better than any of the others. Cruel when necessary but not otherwise, always cognizant of his responsibilities, and understanding that willy-nilly torturing and killing people is merely asking to be overthrown.
The point that I am trying to make is that for him, how Arya incorrectly addresses him is but one of many signs that he has penetrated her disguise. And they show this remarkably well in the show. A peasant would always keep their eyes averted, not speak up and out, not argue or disagree, not do all manner of things that Arya does. Combined with her way of incorrectly addressing him, it is obvious. The disguise works on others because people generally see what they want to see. And the high and powerful rarely see the low and weak, which is frankly one of the many reasons that most of them suck as rulers.
Twyin Lannister, however, is a ruler of a different breed. He actually pays attention to what is around him. Arya is either a peasant trying to appear to have come from noble origins (or something similar) or a noble trying to pretend to be otherwise. Given all the tiny mistakes and slip ups and attitude, the latter is more likely than the former.
This is especially true given that she is in a castle which was overrun, in which case the smart thing for any aristocrat would be to pretend to just be another peasant; generally, if one of the invaders thought she had a touch of noble blood, she might be killed outright just as a safety precaution. Arya is clearly not stupid, and therefore is trying to suppress rather than emphasize her background. She's is imperfect at it, but it functions.
Tywin knows that she isn't what she seems. And it amuses him to keep her as a pet. Had he known exactly who she was, I have little doubt he would have either held her prisoner or killed her without a second thought. But remember, Tywin is only cruel when he feels it necessary. A little girl clearly terrified and powerless who is nonetheless smart enough to keep quiet, maybe comes from a noble family which otherwise no longer exists, is not a threat and is unlikely to ever become one. Thus, there's no point in singling her out for cruelty.
Whereas Tywin finds himself surrounded by people who can't seem to see past the end of their noses. And Tywin, curiously, genuinely deeply loved his wife. Someone like him would seem to be incapable of loving a mindless yes-woman. So he knows and values a female intellect. And having a sharp mind around, that nobody else knows about or even properly really sees, sitting right in his pocket? A useful resource, and oddly a sort of friend for the friendless man. Someone to challenge his wits against, which is probably the closest Tywin gets to having a friend.
So, yes, it seems clear that Tywin knows that Arya isn't just some peasant. And it amuses him to draw her in closer rather than spurn her. If he could gain her trust, at the very least he would effectively create a very loyal spy that nobody else sees who could give him intelligent reports rather than just dumbly reporting conversation. Conversation is all well and good, but even better is someone who sneakily reads documents laying out even upside down while listening to a conversation. Tywin is arranging things such that all roads lead to advantage to him, a more or less trademark trait of his.
Edit: as an aside, it's a very drastic contrast with the book, where Arya serves Lord Bolton rather than Tywin. Bolton is terrifying because of his power and cruelty (flayed man sigil and all that) but is at least reeeeelatively stable (certainly compared to his son). So while he is somewhat unpredictable, so long as Arya is careful and quiet, she is probably relatively safe.
But with Tywin, her every word, every motion, they all betray her and scream what she is, if not who she is. It sets up a very different but I think much more interesting dynamic.
Tywin is not the only one who notices these things, by the way. The Faceless Man does as well. Part of her training of erasing her identity is to eradicate all of these telltale little traces of who and what she is, so hat in the future nobody can be able to detect her via such subtle cues.
Really going off here:
Which of course only further begs the question of how the faceless man ended up in the cage. He can change his looks and his actions at will, and appear to be someone else entirely. The only conclusion I can come to therefore is that he wanted to be captured. Why? Well, we know that people in this world receive messages from their respective gods. What if he was under instruction from his God of death, who wanted Arya as an acolyte for some reason to be revealed later, and this was done as a setup to gather Arya in?
