(This is not meant as personalized tax advice - consult a tax lawyer or accountant for advice on your own tax situation.)
As background, it's helpful to remember why the gift tax exists - it's to close a loophole in the estate tax. If a wealthy parent dies and leaves all their assets to their child, the government collects estate tax on the amount of the estate. So the parent might think to circumvent this by gifting most of their assets to the child before they die. That would deprive the government of a lot of revenue, so the gift tax ensures that such gifts are taxed in a roughly similar way to inheritances.
So yes, if a parent gives their children birthday presents whose value exceeds the annual exclusion, then the parent must report the gift to the IRS (Form 709), and pay gift tax if necessary. Note that they only have to actually pay tax once the total amount of the gifts over their lifetime exceeds a lifetime exclusion amount, currently about $12 million. This tax is really only meant to hit the "1%".
It doesn't matter whether the gifts are of money or other items of value. "The gift tax applies to the transfer by gift of any type of property." If it only applied to money, there'd be another loophole: the parent could dodge the tax by spending all their assets on a $20 million dollar gold bar (or painting, jewelry, rare postage stamp, etc) and give that to the child instead, which the child could then turn around and sell.
The "kiddie tax" is an income tax on children. Gifts are generally not taxable income to the recipient, so the child would not be liable for any income tax, no matter how large the gift or how it is given. (The gift tax is not an income tax and is imposed on the giver, not the recipient.)