Sort of. Since singing probably entails the inclusion of words, maybe sing is the wrong word. Let's say she vocalizes. Could we say she scats? That's probably pushing it.
What she does do is "Ha, Huh, Hum" through "Mary's Lullaby" and "I Lived In Heaven." She does it while she plays, while she eats, while she climbs the stairs...
Unprompted, unled, and on pitch.
And I just think that's cool.
And... nicely unexpected.