We present SIMCO, a completely agnostic multi-class object counting approach. SIMCO starts by detecting foreground objects through a novel Mask RCNN-based architecture trained beforehand (just once) on a brand-new synthetic 2D shape dataset, InShape; the idea is to highlight every object resembling a primitive 2D shape (circle, square, rectangle, etc.). Each object detected is described by a low-dimensional embedding, obtained from a novel similarity-based head branch; this latter implements a triplet loss, encouraging similar objects (same 2D shape + color and scale) to map close. Subsequently, SIMCO uses this embedding for clustering, so that different 'classes' of similar objects can emerge and be counted, making SIMCO the very first multi-class unsupervised counter. The only required assumption is that repeated objects are present in the image. Experiments show that SIMCO provides state-of-the-art scores on counting benchmarks and that it can also help in many challenging image understanding tasks.