As of the Arrow-Canary wedding, yes, everybody and their Goddamn mother knows not only that Oracle used to be Batgirl, but her real name, her address and her favorite flavor of toothpaste. She's always been known as "one of Batman's," though, as that's how she was introduced (and vouched for) onto the League.
That said, why does it even matter if Barry knows who Oracle is? Seriously, it's not like you need to be best buds with someone to do your *job* for them, and Barbara's *job* is doing stuff like that for superheroes. And the whole point of that exchange is that Barry didn't know the League was doing this for him, much less ask them, so where's the need for him personally to have anything to do with it whatsoever? The exchange becomes "Oracle hooked you up," Barry going "wait, who the hell is Oracle," and being told "Man, you really have missed a lot, haven't you." Yet another golden opportunity to drive home how out-of-his-time he is, it fits perfectly.
And unlike saying the former Goddess of Truth built an effective, complex and vigorous lie using her totally nonexistent pull with the US government (even if he missed that that status quo is being dismantled, you'd think he'd have picked up that she was little more than a lackey even when it was up), it actually makes sense for the characters involved.