When designing spacecraft weight is everything - to move something in space you need a proportional amount of fuel, and then you need even more fuel to move that fuel. Tsiolkovsky's equation shows how adding even a small amount of weight to the final stage of a rocket greatly increases the weight of the lower stages. Soviet engineers zero'ed in on one specific element and that was in order to return something to Earth you needed a heatshield, a parachute and other equipment. As a rule of thumb they figured out that for every pound of spacecraft you wanted to bring back to Earth you would add about 2 more pounds to the spacecraft's weight.
Given how much weight was dependent on the size of the return capsule they decided to design it first and make it as small as possible, then build the rest of the ship around whatever they had come up with. The lightest possible return capsule would be a sphere: maximum volume (so you can fit 3 guys) with the minimum mass. But a sphere wouldn't work since it wouldn't remain steady and the G forces would kill everyone. Applying some math from the field of aerodynamics created the 'headlight' shape, providing lift while adding the minimum possible mass. The headlight return capsule is the part that is going to be identical no matter who designs it - the Soviet Union, the American contracters or the Chinese. As long as the design principle of a minimum mass return capsule is used it will look more or less the same from the outside.
The rest of the ship has more room for originality but is still going to be affected by math and common sense. A service module where the engine and fuel go will exist and it will obviously fit at the bottom/base of the spacecraft. To aid in launch aerodynamics it makes sense for this service module to be a cylinder with a rocket on the bottom and sized to fit with the spacecraft's largest surface at the top. Apollo's service module followed the same logic. Finally you need a crew cabin (the orbital module), since the whole point of a longer duration spacecraft is that your guys can get out of their seat. Since the orbital module isn't needed for deorbiting it makes sense for the reentry module to be connected to the service module, and so the orbital module by default gets put on top of the whole stack. Since it has a smaller attachment point anyway (the small end of the reentry module whose shape is already fixed) it might make sense to make the orbital module roughly spherical, since this again maximizes volume : mass and both the Soviet and Chinese versions did that.
General Electric, one of the bidders for the Apollo program, performed a study that came up with a nearly identical craft despite the Soyuz blueprints that existed at the time being a closely guarded Soviet secret. The main difference was their version of the orbital module. Rather then focusing on the volume : mass ratio (sphere) they focused on a shape that would work best for the fairings (Soyuz requires a large fairing to protect it during launch, much like most satellites do). This resulted in a cone shaped orbital module, essentially a lighter more minimal version of the Apollo command module. Of course the GE design was never used because NASA had decided what Apollo would look like long before a million (1960s) taxpayer dollars where spent on the design studies. The NASA design focused on a different key requirement - the module should have the same diameter as the Saturn C-2's upper stage. Because of that requirement the size of the heatshield became a fixed property. With a heatshield that big there was no reason to not bring back the whole spacecraft, minus the service module, and so you got the Apollo design that went to the Moon.