Skye tells him that he has been erased on the phone while he is in Cuba with Garrett, and wonders how he isn't very upset. He says that it isn't the first time his identity has been compromised. It clearly refers to his SHIELD identities, though. As others have pointed out, he has been called Grant by his brother at age 8, and Grant Ward by the detention center at age 16. Not to mention that his name and his family would have to actually exist for him to pass all the background checks before being accepted into SHIELD.
Yeah, that's it. I was going to suggest the quote was in Turn, Turn, Turn, but you might be right that it occurred later during the phone conversation (Providence perhaps?).
Here's the dialogue from "Providence" (apologies for any typos)
SKYE: Hi, it's me. I just wanted to let you know that Grant Ward no longer exists.
WARD: You wanna explain that, Skye?
SKYE: Some trigger happy Colonel threatened to pay The Hub a visit, Coulson had us evacuate before he could lock us all up. Now I'm scrubbing our IDs while Coulson figures out our next move.
WARD: Makes sense. Agent Hand picked up some Navy jets in the vicinity, she's taking the long route to The Fridge, just to be safe.
SKYE: Wow, you took that surprisingly well. Did you hear the part about you no longer existing?
WARD: Well, it's not the first time my identity's been compromised. Any idea where Coulson has us headed?
SKYE: No idea, I'll keep you posted.
WARD: Good, I'll catch up with you when I can.
SKYE: And when I see you next, who ya gonna be? You've got a clean slate, you can be anyone you want.
WARD: I don't know. Maybe Tom Brady. I always wanted to play quarterback.
SKYE: And date supermodels. You are so predictable.
WARD: No, I mean the guy seems to have things pretty worked out.
SKYE: Be careful.
WARD: You too.
GARRETT: Tom Brady?! You hate the Patriots.
WARD: The Agent Grant Ward she knows doesn't.
GARRETT: Yeah, that straight version of you is something else. I don't even think Romanoff could pull that one.
WARD: Well, it wasn't without it's complications.
(they then go on to talk about shooting Skye and all that stuff)
edit: corrected a mistake/typo on one of Skye's lines