Okay, here's the thing I think is awesome. When Tadashi introduces Hiro to baymax, he says, "This card is what makes Baymax Baymax." On Baymax's card (that makes him Baymax), Tadashi wrote his name on it. Baymax had Tadashi in him, which makes his line, "Tadashi is here" so much more deep. When Hiro threw the card, he was becoming like Callahan, throwing the thing he'd lost away from him.