Further to the question, if the onion was native in many places, was this the same species of onion, or did it vary by region in the world?
I asked this same question about garlic, and here was the responses.
Onion is a very old domesticate, prehistoric. By the time we have written sources and artistic sources, onions were widespread through Europe and Asia. Onions don't last as well as cereals, archaeologically, so we don't have any good archaeological evidence of the very early spread of the onion. By 5000 years ago, they were being grown in Mesopotamia, by 4000 years ago in egypt, by 2000 years ago in India and China. These dates are merely the dates of the earliest records that have survived - cultivation there might be much older (and, before cultivation, wild alliums (i.e., onions and their relatives) were widely eaten).
There are many, many wild species - Allium is a very species-rich genus, with species in all continents (apparently even including Antarctica!). Where humans were before history, before civilisation, onions and their relatives were likely to have featured in their cuisines (perhaps greens rather than bulbs). Out of the many hundreds of wild species, only a handful (c. 30) are cultivated as food crops (others are cultivated as ornamentals), and only about half a dozen are widely cultivated. There are four main species of bulb onions: A. cepa, the onion onion (probably domesticated in Central Asia), A. chinense, the Chinese onion, probably domesticated in China, A. fistulosum, the Welsh (i.e., "foreign") onion, probably domesticated in China or nearby in East Asia, A. oschaninii, the French grey shallot (griselle), probably domesticated in Central Asia or western Asia. Wild species such as A. stipitatum, the Persian shallot, are common in some local cuisines but are wild-harvested rather than cultivated. There is also A. canadense, from North America, which was eaten as a wild plant, pre-Columbian, and is today cultivated in Cuba (good references on the history of its cultivation would be appreciated). This diversity of species says that different species were domesticated in different places. We don't know - because it's lost in the mists of prehistory - whether these were domesticated because the idea of growing onions spread, or whether they were domesticated independently on any influence. There was certainly dispersal of cultivated onions, and they might only have been spread north of the Alps in Europe by the Romans (at least, they were popularised north of the Alps by the Romans, along with garlic).
American onions are historically mysterious. Apart from A. canadense which was mentioned above, onions were in the first wave of Old World plants reaching the America with Columbus. A. kunthii was cultivated in Mexico, post-Columbus (and perhaps earlier). Cortez encountered onions being used in Mexico, but the details are a mystery - they could have been wild, they could have been an already-domesticated American species, or Columbus-introduced onions. What is certain is that wild onions were eaten in the America, pre-Columbus; the mystery is about pre-Columbian cultivation and domestication.
I must confess to an ignorance of pre-modern onions in sub-Saharan Africa, and would much appreciate references to such (they are certainly widely used today in sub-Saharan cuisines).
Reading:
For a dedicated book on the history of onions and relatives: Martha Jay, Onions and Garlic : A Global History, Reaktion, 2016.
One of the best detailed technical sources I've seen is: R. M. Fritsch and N. Friesen, "Evolution, Domestication and Taxonomy", chapter 1, pp 5-30, in H. D. Rabinowitch and L. Currah (eds), Allium Crop Science: Recent Advances, CAB International, 2002.
There is also a respectable and useful short entry in the Cambridge World History of Food.
Man, food topics get a lot of karma! You have stumbled on a surprisingly interesting topic!
We don't really know where onions came from. Scholarly consensus places them somewhere in West Asia but the evidence is very weak. The original species they all descended from is dead and we haven't completely cataloged the species. Nor is there big money in the onion business. So we aren't really sure. What we do know is that humans have been eating them since before they were domesticated and cultivated. It's one of the few vegetables (as opposed to cereals) that was planted prior to the development of agriculture.
Let's expand on that a little. Hunter-gatherers will engage in low effort quasi-farming in their territories. Not intensive cultivation of fields but, for example, planting seeds, leaving on their migratory patterns, and then coming back to eat the result. This slightly increases the yield of the area they gather food from. While we're not entirely certain of what triggered the full-scale adoption of agriculture, we do know the crops they first farmed were first subjected to this quasi-farming. Before intensive cultivation, hunter-gatherers would scatter proto-wheat seeds in fields, leave, and come back later to harvest them.
Onions are attractive in this model for several reasons. Onions are flavorful. Their pungent odor and growth beneath the earth make them less likely to be eaten by herbivores. In fact, they're toxic to many animals. They are hardy plants that last a long time after harvest. They are light and easy to store. They also require no preparation: you can eat an onion like an apple (and they are even wrapped in a handy skin container). They produce easy to plant seeds. And they are very uncomplaining plants: they will grow in bad soil and can be planted in most places. On top of that, if you leave them in the field they don't rot (like wheat). They can even last years after the initial planting. All this at the cost of some bad breath. (And we imagine our prehistoric ancestors probably didn't smell very nice anyway...)
We know humans were eating onions by the sixth millennium BC in the Middle East. At the latest. There's evidence they started before that.
Humans brought onions with them as they migrated or found and cultivated local versions. Again, I cannot stress this enough: onions are uncomplaining and versatile. If you were a Mongol herder, you could throw them down and let them grow then bring your herds back to eat them. If you were a Papua New Guinean tribesman, you could cultivate them in small gardens or throw the seeds down to suit your hunter-gatherer lifestyle. If you were a farmer in England you could grow them on the land that wasn't good enough for wheat and use them to flavor stew or feed any animals you were fortunate enough to own. It didn't matter if you were on the frozen steppe, temperate England, or Papua New Guinea: the local version of the onion would grow.
We know of no major culture that hasn't had some form of shallot, onion or pseudo-onion in historical times. In fact, there were at least two attempts to 'introduce' onions to the New World during the Colombian exchange. In both cases, the natives recognized the onions as similar to the plant they cultivated.
So, yes, this has been the case for a certain value of 'always'. This has been the case since before recorded history. (The onion is so old it has an oft-overlooked mythological history, by the way. The Egyptians sometimes buried pharoahs with onions, for example.) But at the time many cultures had their own individual onions. There are at least three hundred different types of onions and may be as many as a thousand. But only three basic types make up the vast majority (somewhere in the neighborhood of 90%) of world production. The imaginatively named Yellow, Red, and White Onions. Why only the three? Commercial farming.
But onions were relatively slow to marketize. As late as the 17th century, the King of England's kitchen had trouble making up an onion shortfall because they weren't being sold in London. It wasn't that no one had them, it was just that they weren't at market. They weren't worth enough or in high demand. People would grow them in their own homes if they wanted them. Once the kitchen went directly to some farmers they were able to buy the onions (and cheaply). The problem with being very easy to grow on even very marginal land is that everyone will get in on the game.
So how did onion farming become business?
America and the Market Revolution.
The United States' economy, both before and directly after the Revolution, focused on the export of primary goods or basic finished goods to more advanced economies and to other resource extraction economies like the sugar plantations. The United States, by dint of its proximity to the Caribbean and sugar colonies, had long exported foods that would not otherwise be economical to export. The sugar colonies grew almost no food so they had to import everything, even things that could be easily grown elsewhere. Every inch of arable soil in places like Barbados was put to sugar production.
The United States had several agricultural systems at first. We are most familiar with the plantation and small farmer system because they are the ones that survived until 1862 (and went to war with each other). But there were others. New York was originally a manorial economy. There were communal systems of farming in Pennsylvania and parts of New England. There was even a commons system in some parts of Vermont and New Hampshire. For complex reasons having to do with cash profitability, incentive systems, and the way these methods affected the marginal value of labor, the small farmer system encouraged marketization. It also helped that the North, where the small farmer system grew up, was more communal and urban.
By the 18th century (the early stages of the Market Revolution), dedicated onion farms begin to pop up to service this growing export market among these small farmers. They were stereotyped as poor because onions were cheap and they lived on marginal land. In reality, onion farms were fairly profitable. Onions have never fetched a high price but they are easy to grow so at scale they could be very profitable. Plus, the land they needed was often cheap, lowering capital requirements for an onion farm. New England was particularly notable and Connecticut used to be the onion capital of the entire world (or, more realistically, the Americas and Europe). It was on these large farms that standardized onions were produced, at first reds, then later yellows (which grew especially popular during the westward expansion). Whites came later and are more bound up in the history of Latin America. Although America did gain, among other things, several white onion farms.
The creation of a market for onions encouraged others to enter and compete. The Russians, for example, had a great deal of cheap labor and marginal soil and entered early. They often imported massive amounts of seed from the United States, leading to a common number of strains. And the cheap mass produced onions (as cheap mass production tends to do) pushed out producers on marginal land, except the occasional garden or people farming local varieties.
As the US increasingly became a capital-rich nation its farming became increasingly capital intensive and many inventions were invented to increase yield. But onion farming can be made as a labor-intensive crop, so onion farming shifted to places with cheaper labor. The US retained only the part of the market sustainable by high capital investments. Since then, the market has shifted around, following cheap labor. Right now, India and China (with their deep and cheap labor pools) have come to dominate the onion market. As much as anyone can. As an easy crop with low barriers to entry, it's difficult to maintain a consistent hold on the market. Even China and India are starting to be undercut by African countries like Kenya.
(Fun fact: The US banned market speculation in onions in 1952. The ban is still in effect.)
TL;DR: Onions have been around since prehistoric times in almost all cultures but they are different varieties of onions, though more than 90% of all onions produced today are one of three types.
Sources
Garlic and Other Alliums, Innovations and Inventions in U.S. Onion Farming, A History of the Onion, The First Farmers, and The Origins of Agriculture.