Tell Your Kid That Infinity Comes in Different Sizes

Good try, and there are indeed different types of infinities, but your examples both represent the same "countable" infinity of the natural numbers! You can create a bijection between these two sets. I can't see anything helpful about this example for kids, and in fact I think it teaches them the wrong intuition.

If you wanted to show different infinities, there might be a way to make Cantor's diagonal argument more accessible to children, to show them countable vs uncountable infinities, but that's a lot less intuitive!

What I mean by a bijection: in your example, you can map the nth dog to the nth paw, and each dog will always have a unique corresponding paw (albeit not its own paw, except for the first dog). Showing them different infinities would mean showing two sets where it's not possible to have this kind of mapping.

For example, an integer numbering to the set of all possible dog names (of unbounded length)

Or leave them alone and let them realize that mathematicians who believe this nonsense are silly.

The number of even numbers = ∞

The number of natural numbers = ∞

The larger consensus mathematicians seems to be that these two infinities are different... which shows that most mathematicians have lost track of what infinity is supposed to represent. They define it as boundless but want to desperately slip it as a bounded quantity for their amusement and mental gymnastics like the Hilbert Hotel or the Banach-Tarski.

I would love to believe that they are confusing kids now because the adults have stopped falling for this nonsense... but I don't think that is the case here.