When you stretch a chain the distance between pegs become longer than designed. Even fatal chain stretch cannot be easily identified by looking at the chain.
There are very simple yet accurate chain scales used to identify whether the stretch is none (new chain), fair, acceptable, "replace the chain", "replace all chain and sprockets". Another way to estimate the chain stretch is to try to lift the chain from the sprocket. The further you can lift it, the more stretched chain you have.
One can see the chain is stretched far above the "replace the chain ASAP" level when the teeth on small sprockets are no more symmetrical and become worn from the rear side (upper teeth of the rear sprocket). When the chain starts skipping over the teeth, whole drivetrain it worn out and the chain still looks quite good to many people and is still very strong in tension and the peg-to-peg distance is stetched in orders of tenths of milimeters.
Another reason for not-stretching a chain is the momentum one can produce on such bike. The crank arm is shorter on kid bikes than on the "full size" bikes. Therefore the tensile force you have applied to the chain was lower than you are able to apply on your bike.
Since you were able to load the chain with weaker forces and even hard stretch will be negligible to wrong setup the only option left is that you have moved the rear wheel a bit front. Realign it, as others already suggested, and tighten it.
It is worth nothing to get the chain measured for the wear, though.