Baby-talk alteration of blanket. By surface analysis, clipping of blanket + -y.
Meanings
A child's blanket.
His banky, the grubby woollen blanket that he carried with him everywhere, was draped over the rest of his bedclothes. 2013, Kate Thompson, The Missing Link, page 6:
You sound like a little kid wrapped up in his banky watching his favorite t.v. show. 1998 July 15, Logan7, “Moe's A Joke”, in alt.fan.bob-larson (Usenet):