It's the babysitter effect. It costs a whole lot more to ensure a safe, and healthy environment for kids. And let's not forget about fun.
Why is kids Chess double the price of chess.com?

It makes no sense to have it twice the price.

I think Chesskid may involve more masters and coaches than the main site. A lot of kiddie videos and stuff, and interactive lessons. Who knows.

It makes no sense to have it twice the price.
It is a completely different site, with different staff, different features and formats.

It makes no sense to have it twice the price.
It is a completely different site, with different staff, different features and formats.
Nope.
Same developers:
Same engine...

It makes no sense to have it twice the price.
It is a completely different site, with different staff, different features and formats.
Nope.
Same developers:
Same engine...
Absolutely the same company but different everything else; they have their own site, own development team for both the web and apps, own support staff, etc. They are separate platforms, have separate targeted audiences, and ChessKid.com has a different design paradigm for running school clubs and coaching, with the tools needed to support that and make sure kids are only involved with coaches and clubs allowed by their parents/guardians. It is very likely they host out of the same datacenters.

Ok I understand... still doesn't explain the price.
You have the Gold membership, which is about a third of the price of the Diamond membership, which has access to all the lessons and videos.
I don't know Chesskid but I assume they have videos and lessons.
So it would be fairer to compare Chesskid price to the Diamond membership at $99 and then ask why Chesskid is so cheap?
Why is the kids app twice the price of the normal app?