Canada here, slight differences.
For my kids I have a savings account set up for each that contains birthday money, gifts and oddball amounts. The accounts have under $1000 and generate modest amounts of interest. As they get older I'll switch them to GIC's and other stable products and use them as an example of real life investing. I find math is easier to learn when dealing with real money and not hypotheticals, even as a kid. The GIC's are because as a kid I don't want them turned off investing in a bad year; I'm not sure how they'll react to a 20% stock drop at age 12. In Canada there's benefits to having kids earn income but no drawbacks, later on my kids will have a small amount of extra RRSP (401K) room, it rolls over here and is based upon annual earnings.
My first "investment" was a savings bond my Grandma gave me. It was $200 and worth more in education than any money I earned from it. Childhood savings should also focus on education, not just returns.
I have a separate education account (RESP - similar to 529) with my money going in to fund their school. If its not used for school I would have been better off having a private investment account set up in their name. The key is to have their accounts attributed to them so any earnings are taxed in their hands and not mine. I can control the accounts, I don't want to be paying taxes on their money though.