Unlike Bruce, Natasha is proud of the fact that she is the Black Widow. She says it like that because it's like the Black Widow is an alter ego or something, because the Widow would never find herself using sarcasm or deadpan looks or anything but violence no matter who she was talking to, whether it be Clint or Thor.
She has been trained to be better than that.
But as Natasha Romanoff she teases her team mates and throws Tony out of the lab and spars with Clint, as well as trying to help Steve get over his mental pictures of women being less durable on the front lines. She thinks that, maybe, she's managed it.
The Black Widow is what she becomes to defend them, something dark and inhuman, with no proper capacity for human feelings. She is merciless.
Because she knows that is how it has to be, to keep the very few she cares about safe and to wipe the red off her ledger.
She is Natasha Romanoff and the Black Widow and she knows how much she needs both of them to survive.