You can pretty much put any carb on any moped as long as it physically fits under/next to the frame, and you have an intake manifold that works with the engine and the carb, and the carb size isn't totally different from what the motor was designed for. Most aftermarket intakes are designed for bigger carbs.
Anyway...to understand the whole carb landscape, it helps to understand geography. Most stock carbs on mopeds tend to be of the same country (or at least region) as the moped itself.
This isn't a hard and fast rule, but I've found it to be the case most of the time. There are also some other moped and carb brands but the big 4 above cover probably 95% of mopeds in the western world. Sometimes there are slightly bigger sizes of stock carbs you can get as a modest upgrade (i.e. going from a Dellorto SHA 12mm to a 14mm, or going from a 14mm Bing to a 17mm Bing).
If you see a big (18mm+) carb on a moped rally in N. America, chances are it will be a:
Speaking of clones...at this point there are so many Chinese clones of original carb designs (especially of Keihin carbs, because these were what Honda used on engine designs now universally cloned in China) that people often speak of Asian carbs in terms of their overall type (like CVK or PWK in the case of Keihin) as often as they talk about the original manufacturer, although this is more typical of the scooter and underbone worlds.
There are also some Indian carb clones of Bing and Dellorto carbs (like Pacco and Spaco) since Italian motors and their clones by Bajaj, LML, etc. became popular in south Asia, analogous to Honda designs in China.
That's as broad of a carb overview as it pertains to mopeds as I can think of right now; anything beyond that is probably specific to a certain type.