Would it turn you off if someone you started talking to asked how your kid is doing instead of saying daughter?

I know a guy who will ask for a tablecloth when they mean a napkin.

I know it's not the same thing but
The fact he is referencing your daughter is something. I get the impression you want him to show his depth of genuine interest but this choice of word isn't offhand enough to be evidence of anything, in my opinion anyway

If he avoids being precise in lots of other ways, then okay that's a potential issue.

If it's early days, the fact he asked about you and your child looks positive.

Not at all. It’s nice that he’s asking.