For the record, by “your kids” I don’t just mean your offspring. It could include your nieces/nephews, kids you look after etc. Just kids you saw often enough to have a pretty good sense of the answer. A lot of childless people probably have valuable insight, so I’m not going to leave them out 
I’ve worked with a lot of children and some infants over the years, but very few toddlers much older than one and less than three. None of the one-year-olds I knew used their own names to refer to themselves (though they’d look if you said their names, so they knew them) and all the three-year-olds would offer their names if asked. So I assume the use of one’s name typically begins sometime after age one but before age three.
So how old were the kids you know well when they began to label themselves with their own names? And if you know, was it before or after they began to label their parents purposefully? (after “mama” actually meant her specifically, not just as one of their few words) My guess is that they label themselves after their parents, but we’ll see how that shakes out.