I bought some shorter Tapcons and noticed the shank on the new ones was thicker than the old ones I had. The short ones went in as easy as could be expected but the shank did not break as the old ones did. 1/4" Tapcons with 3/16" drilled hole as they suggest. They must have made them thicker due to breaking.
Is your wall made of cake? That first picture looks like cake.
But seriously, look at the plastic anchor your tried to put in that hole and compare it to another one. I think you will find that the used one is broken. What happened is you screwed that anchor in to a stud, and it broke, leaving plastic behind.
Drywall anchors are used to hang things on ...