No one actually says "my child is 3" meaning anything other than 3 years. They would say "3 days", "3 weeks" or "3 months" meaning the other lengths of time.

Not where I live. You'd never specify how old somebody is with a bare number and have it mean anything other than years in the US. "My kid is 3" is always 3 years. So is "How old is your Tammy?" "Three". That only ever means years. Every other unit is always explicit. In my decades as a parent and being around other parents of kids and newborns, I've never experienced an exception to this.

That's the point. If you are talking colloquially to someone where you live, sure, whatever, this doesn't matter. But if you are writing something down in some publication for a wide and unpredictable audience, you should use units.

Most of the time if someone says "he's 3", it is a good bet that they mean 3 years. People usually specify if they mean days/weeks/months with respect to someone's age. Not always, of course, but it's definitely uncommon to drop the unit when it's anything except years.