I think there's another angle to consider this question from, but unfortunately it's not all raw history. Additionally, Sengoku Japan isn't something I have particular expertise on -- but I think there's something there that links the "true" history and the "cinematic" histories that have previously been suggested in this thread. So I leave this post at the mercy of the mods.

It has previously been suggested within this thread that the comparison between the two periods in question is derived from cinematic influence -- that is, mid-20th Japanese filmmakers imitating aspects of popular American Westerns. This is undoubtedly true; consider how even Akira Kurosawa was occasionally criticised for appealing to the West, despite how distinctly his films are tied to Japanese identity and concerns. While Kurosawa's films (and the films of many of his contemporaries) did contain elements of Western influence, they were unique.

Those elements of Western influence? Remember that the mid-20th century Japanese cinematic context was partially informed by, in particular, American Westerns. That kind of film is overwhelmingly centred around the romantic individualist, operating outside the law, where no society can contain them -- in other words, a picture-perfect view of the American Protestant ideal. Such characters were self-sufficient, self-motivated, and owed allegiance to nothing but their moral codes. The cultural stereotype of a cowboy remains that of a rough and raw individual, who nevertheless has a heart of gold.

You might imagine that this character archetype has little place in the more collectivist perspectives prevalent among a society like the Japanese. That's almost true. But it doesn't account for the ronin, usually translated into English as "vagrant", and usually referring to a samurai without a master. The term goes back further, though; its initial usage was in reference to a peasant or commoner without a master (I believe this is something like a 12th-13th century usage), but during the Sengoku period, it was more common to use the term to describe samurai (or even just any kind of warrior) without a leader. In any period of Japanese history, it has been shameful to be a ronin, to the extent that the term is still used to refer derogatorily to people who fall through the cracks in society, or even to those who are between jobs (loyalty is considered a big part of corporate trust in Japan).

You can see the similarities between the archetype of the cowboy and the archetype of the ronin. Both live unstable, nomadic existences in settled societies, often outside the usual social structure, and are distrusted by the people they ultimately serve. The difference is that American cinematic culture was predisposed towards a romantic depiction of this character archetype. Unlike a largely Protestant American population, the Japanese identity was not based on gaining both political and religious independence from another power. The American view of individualism was a likely contributor towards what would become the commonality of the ronin as a central protagonist in Japanese historical and fantasy media.

Ronin had been romanticised before. The story of the 47 Ronin, recently adapted into a terrible, orientalist mess, was a well-known story of honour and revenge. But that story is about those ronin regaining honour through commitment to their deceased lord and the conventional social structure -- those ronin commit ritual suicide at the end of the tale, ergo the actual grave of the forty-seven in Japan. I say "tale", because while there's plenty of evidence to suggest that the story of the 47 Ronin is based on truth, it could have been subject to biased interpretations and revisions in the act of writing.

The new ronin characters broke from the standard set by the tale of the 47 Ronin. While those ronin were characterised by their commitment to the conventional social structure, the new popular depiction of a ronin became one who was accepting of their circumstances, living the life of a ronin as best they could. Doubtless, the history and mythology of Miyamoto Mushashi also played into this, himself a Sengoku-era master swordsman and ronin. Consider that one of Japan's most ubiquitously venerated historical figures was a ronin for much of his life and the influence of cowboy individualism, and you get Japan's closest cultural equivalent.

So yes, I think the cinematic influence is the biggest thread that links Sengoku Japan and the American Wild West, as character archetypes from those periods shared common themes that were drawn upon for 20th century popular entertainment. The periods were not necessarily similar in terms of raw, known information, but certain kinds of people lead similar lives, and both make excellent protagonists for action entertainment media.

Pitched battles and open civil war in the Sengoku period. Occasional violent clashes between competing commercial interests in the wild west. Destruction of entire cities in the Sengoku period. Founding of new cities in gold and cattle booms. Competition for limited resources vs competition to control apparently limitless resources. Rise in power of elite warrior class vs rise of middle class farmers and merchants. Rise of elite warrior class vs the great equalizer of multiple shot firearms. Famine due to war vs abundant food with new developments in agriculture.

There is practically no similarity between the two. The only similarity lies in the legend of great warriors. In the Sengoku period you would have Samurai in the Wild West you would have gunfighters. These two are very similar in that they never existed in the way they are written about in popular fiction.

I would say that the Sengoku jidai and American Wild West are a bit similar in that they are points in history that are frequently romanticized both in Japan and America, respectively, as well as in other places around the world thanks to the international popularity of American and Japanese culture. The wild west conjures up images of cowboys, sheriffs, natives, et cetera, whilst the Sengoku period is where we get most of are stereotypes of samurai and ninja from. Indeed, shinobi or ninja are mostly a creation of the Sengoku period, whilst our idea of who samurai are and what they did reflects the era. Samurai were definitely around for centuries before the Sengoku period, but most samurai in popular culture are closer to the ones that were around in the 16th century than, say, the samurai of the 12th century Genpei War, the conflict that is the focus of the epic Heike Monogatari. The wild west and the Sengoku period are periods in history that lend themselves to stories featuring romance and violence, and as such figure prominently in pop culture and national identity.

In terms of the actual daily lives of the eras themselves, however, they are quite different. The Sengoku period resembles Europe in the High Middle Ages more than it resembles any point in American history. Indeed, people often talk about "medieval" or "feudal" Japan, and while there are some important differences between feudalism in Japan and feudalism in Europe (just as how feudalism in Europe itself differed from nation to nation), in general, the similarities are pretty striking. The Sengoku jidai involved a dissolution of centralized political power in Japan. Contrary to popular belief, there were actually shoguns for most of the period - the Ashikaga shogunate based in Kyoto lasted all the way until it was dissolved by the great warlord Oda Nobunaga in 1573 - but these shoguns had little power and the daimyo, or feudal lords, often ignored their authority. The Sengoku period did not involve a clear-cut conflict such as the earlier Genpai War or Northern and Southern Courts period, but instead, featured desultory conflicts between feudal lords attempting to survive or build their own power bases, conflicts that only ended with the unification of Japan under Tokugawa Ieyasu.

I suppose what I am trying to iterate is that the Sengoku period was a period of warfare. Not constant warfare, and not universal warfare, but the period is littered with sieges, battles, assassinations, political intrigue, et cetera. This impacted not only the samurai and the daimyo, but also the peasants who served in their armies and made up the backbone of their power - wealth in Japan was tied to agricultural output, after all. We aren't quite talking about a level of violence akin to the 30 Year's War that would occur in Europe shortly after the Sengoku period, but we are still talking about sieges, sackings, pillaging, et cetera.

Now, I am no expert on the wild west, but if I understand it correctly, the general impression of that point in time is that it was a lot less "wild" than it is commonly made out to be. Events such as shoot-outs or carriage robberies occurred, but not with the frequency that Hollywood might have one believe. Many places in the wild west were pretty much safe, by the standards of the day. This isn't true if you happened to be a Native American, perhaps, but in general there was more government authority wild west than people often believe.

In addition to the other answers give here, part of the reason for making this comparison is less to do with the actual periods themselves than their artistic presentations within popular culture, specifically cinema and later television. Many of the filmmakers specializing in Jidaigeki genre films between the 1930-50s were heavily influenced by the Westerns produced by Hollywood during the 1930s and 40s such as John Ford's Stagecoach. Although the Japanese filmmakers picked up on certain themes of Westerns, part of their appeal was also aesthetic. Many Japanese directors and cinematographers were drawn to the way in which directors like Ford staged their movies and how they conveyed action through cinematography. As Japanese filmmakers like Ozu and Kurosawa became more well-known, their cinematic techniques filtered into those of the later generation of Western filmmakers of Westerns like those of Sergio Leone.

This is not to say that historical comparison between these two eras is misplaced (far from it), but it is highly advisable to keep this cross-cultural context in mind. These cinematic norms and tropes have permeated much of the popular consciousness (lawlessness, the heroic warrior, etc.) that they impel comparisons, even when they when they can be less than apt.