To my ear, this is an error

I just spoke to someone with very fluent English, but a heavy accent and she referred to her child, the subject of a custody dispute, as "the kid." "He picked up the kid."

I know people probably learn that "kid" is a term interchangeable with "child." And it is. But in this emotional context, it sounds - off. I suggest, at the very least, "my kid." But better still - "my little girl." Or "my baby." The word kid, to me, connotes to me something casual, maybe even ever-so-slightly pejorative - the kids next door, those kids at the school. They are not *your* kids. They might even be up to no good.

"My kid" seems more like something you say when you are bragging - my kid scored the winning goal! It's kind of objectifying - distances the parent from the child. That kid is no momma's boy/girl!

If you are really worried, especially about a toddler or small child, kid isn't personal enough.

Anyway, that's just how I feel about it...