The source code is available if you would like to check it out:
https://github.com/lcapps-luke/Takogochi/tree/main/source
It uses the "Nape" physics system. The Takos are made from small circular physics objects (named "nodes" in the source) that are connected with "WeldJoint"s acting as springs. See "PhysicsMeshSprite.hx" for the main part of this.
Nodes of the Takos that are close to the mouse are selected when you press the mouse button. Nodes that are selected are continuously accelerated towards the mouse. This is all done in "PlayState.hx" in the "grab()" and "updateHold()" methods.