I'll try to answer this from an architecture history perspective, since I'm an architect.

Since what we consider the dawn of human civilization in the late neolithic period, humans have stopped seeking shelter in natural constructs and decided to start building their own - eventually nomadic tribes settled down, and we know how the story goes.

At first, there was no industrial refining of materials, so early civilizations had to use raw materials they found in nature - primarily stone, wood and clay (as a material for making bricks). Mortar as a binding material would also be invented later. Let's ignore things like material availability for now, but that was also an important element.

Out of these 3 basic materials, wood stands out - because it grows organically, so you can't just cut out a bigger block of wood from your wood quarry - you're limited by the size of the tree trunk. Also, without proper protection, wood is easily worn down by atmospheric elements, mainly precipitation - as such using it for something as big and intended as permanent as important buildings, be they the mesoamerican, egyptian or mesopotamian is out of the question. At this stage of human development, wood was probably used for shacks and similar cheap habitation architecture - since it was not protected, we have little to none archaeological evidence about this.

So we come to the two other choices - clay (brick) and stone. While the mesopotamian ziggurats were made of clay, egyptian and mesoamerican pyramids were made of stone. Worth mentioning, one thing both clay and stone have in common compared is their excellent compressive strength - offset by their practically nonexistent tensile strength. While wood's compressive and tensile strength are more or less similar, stone and clay building elements break easily when faced with larger construction spans that cause greater tensile forces - and require a larger section to compensate for it. Early civilizations understood this; just looking at a layout of a hypostyle hall in an egyptian temple complex like Luxor or Karnak, you understand its builders were aware of the limits of the stone as a building material. The spans are minimal, the columns are huge. The lack of mortar or any kind of binding material didn't help. It would have to be a few thousand years until Romans would solve this issue by inventing the architectural element of the arch - a form which can span over larger distances, with all of its elements only being under compressive pressure, and again thousand of years later when we invented reinforced concrete - where steel rebars take over the tensile forces, while concrete itself takes over compressive ones.

Let's talk about the egyptian civilization for a bit, because in their case, we can see a clear evolution of the pyramid form - Egyptians first buried their (important) dead in mastabas - which are simply stone or clay mounds. As the civilization developed, they were keen to develop this concept and make greater buildings to stand the test of time. So they started stacking mastabas on top of each other, giving us the early terraced pyramids, that later evolved into more aesthetically pleasing, geometric pyramids. Some pyramids' slopes even changed their inclination mid construction, like Sneferu's so called Bent Pyramid at Dahshur, which gives us a nice insight on how their architectural ideas developed, sometimes even in situ.

But back to raw material physics - let's say you're placing 4 stone blocks on top of each other - each block weighing a ton. Keep in mind that in loadbearing physics compressive forces on one part can easily translate into tensile forces into the other part of the material. The highest stone block only has its own weight to carry. The one below him, 1 ton. The one further down, 2 tons. The further down we go, the compressing force is greater, and also, tensile forces that are minimal and negligible at the top stone block become greater ( this is one of the reason Gothic architecture uses flying butresses - they take over tensile forces inflicted by the weight of the high vault ) Then account external horizontal forces like wind, and you'd realize that a stone cube made in a similar manner simply wouldn't stand - the instability of the edges where tensile momentums are the greatest would simply cause it to, best case scenario, lose the aforementioned edges, worst case scenario, topple alltogether. With a pyramid, the higher you go, the less weight the building has, because there's simply less of it. After all, ancent civilizations probably didn't invent the pyramid shape ab nihil- they probably noticed how hills and mountains were shaped and copied that concept.

So to summarize, there's nothing weird with all ancient cultures opting for the more or less similar concept - when the only 'permanent' material has no tensile strength whatsoever but has a lot of weight, you don't have complex architectural elements like the arch and flying buttresses, and you're trying to build something big - the pyramid, or the terraced pyramid is the most sensible and logical architectural solution.

Sources: Vladimir Bedenko, History of Architecture I, 2006