Bucket pulley/hoist for kids climbing frame
What you are looking for is called a "manual chain hoist" and if you search for that you can find some light weight low cost versions from several sources. But you have to understand that these are not toys and little fingers can get injured. To keep from dropping the load if you let go of the chain, they will have some sort of braking mechanism; either a pawl and ratchet, or a friction clutch. The ratchet and pawl type come with a risk of pinch points and someone releasing the ratchet with too much of a load, letting it free-fall. The one in your photo example looks like it is probably a friction clutch version. The problem with those is that they wear and need periodic adjustment, so neglecting that can result in the load falling and hurting someone. That playground likely has an assigned caretaker who's job it is to measure the holding force and adjust the tension on a regular basis.

How about just tying a rope to something at the top and your kid can pull it up? it would inherently limit the weight and thereby the danger of it falling.