How can I create a clay (in a Play-Doh style) material like the one in the image below?
Something like this might be a good start.
Two Voronoi textures are distorted by two noise textures, each Voronoi texture controls a color ramp with a few different colors.
The results are mixed together and get even more distorted by another noise texture. The Mapping node at the start controls the general direction of the clay strains.
Link to the .blend file:
You can play with all the values, maybe you get even closer to what you want.
3d model the basic design, then subdivide in Edit mode then sculpt and click any options (clay is recommended) to give it smooth shading. If you don't want square holes, then just use the Boolean modifier to make a hole using a cone or a cylinder, then texture it with vertex paint or UV unwrap it with the image that you have and you're done!
P.S.: I made a video in it I can send it to you if you want.