Property of Bread Dough
The video doesn't really show much of the dough or how it is handled, I only see a prepared square sheet being filled. But if you actually want dough which handles like play-dough, then the property is called not elasticity, but opposite, plasticity - it is a material which can be formed without springing back. 

If that is what you want, for a bread dough, you can achieve it by 
- reducing the hydration 
- adding more butter 
- using a flour with little gluten content 
- not using any prolonged rises or techniques like stretch and fold, just doing a short rise, without getting it to go too voluminous 

But bread dough is not really a good candidate for that, because it is usually elastic by nature. If you have a need for plastic dough, the best option would be to make a pasta dough with eggs and AP flour. It is plastic like no other. Or also salt dough for ornaments, it is not only a better play doh substitute, and keeps its shape after baking.