The Twintail Monster was an NPC that used to be found in Heart World, where you could interact with it to receive the Twintail effect in previous versions of Yume 2kki. Now it is well hidden in the Broken Faces Area.
To find it, enter Geometry World and find the large scribble on the ground, located north of the entrance. Interact with the left side of it to be transported to an area with a face smoking a cigar (known as "Smokin' HAL"). If his face is white, walk into his mouth and you'll come back out again, but this time the background will be black, his head will be blue, and his eyes will be bloodshot. Walk off the end of the cigar and you'll be transported to the Broken Faces Area (if his head is already blue then just walk to the left like usual). Once in the Broken Faces Area, locate the area where the Marginal Vivid Worker (the thing you interact with to get the Marginal effect) and Surimuki are. Use the Glasses effect near them and the Twintail Monster will appear.
Judging by the mark on the back of its head, it could be Surimuki's head.