From an older answer:

The style was called "toothbrush mustache" or after its other famous adopter the "Charlie Chaplin". It was first pioneered in the United States as part of a general trend of flamboyant mustaches in the late 19th century. Elaborating on the subject of late 19th century mustaches and their relation to how masculinity was performed, Christopher Oldstone-Moore writes in his article Mustaches and Masculine Codes in Early Twnetieth-Century America:

The dignity of beards to which the writer [of the Harper's Weekly 1903 article "The Passing of Beards"] referred was the fading ideal of the late nineteenth century, when full beards had become quite respectable in Europe and America. In a time of rapid economic and social change, men grew beards or mustaches to underscore the natural virtues, and supposedly imprescriptible rights, of men. Mustaches, as opposed to beards, were strongly associated with military fierceness and discipline because most European armies required them for officers, and in some cases, enlisted men as well. According to research conducted by the British military in 1913, the origins of the military mustache were to be found in the style of the Croatian hussars of the Austrian imperial forces, who, in the words of the study, "evidently considered that a fierce mustache would assist in terrifying their enemies much in the same way as Chinese soldiers adopted awe-in of their uniform." Such impressive fierceness and dash was even emulated by Napoleon's cavalrymen, which helped to establish this remainder of the century. When mustaches became wide civilian men in the 1880s and 90s, these men inevitably acquired for themselves something of a "dashing" martial air.

So while the association of mustaches and beards with masculinity generally faded in early 20th century America – some say in part thanks to Gilette's invention of the safety razor – and was replaced with the generally association of a clean shaven face as an honest face, it was still around the same time that two of the most famous wearers of the toothbrush mustache adopted the style. Incidentally, Chaplin and Hitler were born in the same year, 1889, and therefore of a similar generation that had most likely grown up around men who did perform their masculinity in line with the trend at the time, meaning by wearing either fully grown beards or some form of mustache.

Chaplin adopted the mustache sometime around 1914. He claimed in an interview that the idea behind it was to add to the comical appearance of his Tramp character without obscuring his face and his facial expressions too much; something rather important for a silent film actor. While Chaplin's adoption of a mustache in times when the association of beards and mustaches with "dashing" masculinity was fading around America could be seen as an attempt of humorous subversion of tropes of masculinity, Oldstone-Moore also cautions that Hollywood did remain a bastion of oftentimes eccentric facial hair well into the 1930s:

In early twentieth-century America, there was one arena besides politics in which in which the social role of mustaches was especially important, and that was Hollywood, the American cultural stage par excellence. In this landscape of the imagination, an idealized masculinity was enacted in fantasies of power and wish-fulfillment that provided a satisfying escape from ordinary life. In Hollywood, the transgressive was possible, particularly the rugged and rakish individualism of larger-than-life heroes. It is no wonder, then, that one found a much greater proportion of Hollywood leading men with atypical facial hair. In the 1920s and even more so in the 1930s, mustaches became the hallmark of swashbuckling romantic heroes such as Clark Gable, Errol Flynn and Douglas Fairbanks, father and son. On the other extreme, it was also the hallmark of Charlie Chaplin, whose small toothbrush mustache indicated the comic- pathetic opposite of the leading man. The symbolic import of mustaches was underscored by the minute attention paid to them in the Hollywood press. Every appearance, disappearance or alteration of the stars' mustaches generated extensive commentary. In both cinematic and political image-making, the mustache played a remarkably similar role in defining contrasting masculine virtues.

As for Hitler, who was neither American nor a Hollywood actor, various theories have been proposed of how he came to adopt the same style mustache as Chaplin with the only thing we can be sure of being that he did so later than Chaplin and 1914. Sporting the other style of mustache that was popular in Germany at the time, the Kaiser Mustache (Hitler being the one on the left with the "X" over his head) during his time as a soldier, Ron Rosenbaum mentions that while there is no evidence there is still heavily speculation that Hitler modeled his mustache after Charlie Chaplin. Others have proposed the theory that Hitler had to shave his mustache because of German army regulations regarding gas masks.

As for his continued choice to wear this style of beard, it is noticeable that in the Weimar Republic the generally trend of the clean shaven face as a marker of male honesty that was around in the US did not take hold. From Friedrich Ebert, first president of Germany to other politicans that would become symbols of Weimar Democracy such as Gustav Stresemann and Walter Rathenau, the mustache and more seldom, beard, was a trademark of the Weimar politician with only Ernst Thälmann who did have little hair in the first place being a noticeable exception.

Having established such a noticeable trademark in terms of facial hair, Hitler also kept his mustache through times when the popularity of facial hair was waning in general in the 1930s. Most other public representatives of the Nazi movement such as Göring, Hess, Goebbels etc. had either never embraced facial hair or eschewed it in the 1930s but for Hitler it stayed becoming almost something of a trademark.

And speaking from a cultural historical perspective this makes perfect sense: To further reiterate a point by Oldstone-Moore, facial hair is perhaps the strongest fashion statement associated with masculinity because while women can wear suits e.g. (and did so in the 1920s), having authentic facial hair is something that is usually not possible for women but only a fashion option for men. At the same time as he points out facial hair "has been recognized as a choice mask intended either to disguise flaws or assert a character type. Jospeh Schusser, for example, who was the manager of a chain of a chain of New York City barber shops, observed in a 1923 interview that "the most interesting fact about beards and even mustaches" is that "they help to provide a mask, behind which a man can take shelter."

In line with that and the general assertion that beards and mustaches are a physical performance of masculinity, these aspects – being mask, trademark, and performance of masculinity – also cross over in the pop cultural and political arena. Like for Chaplin, for Hitler the toothbrush mustache was a trademark, a public signifier of recognition. Hence Hitler was able to print posters for the 1932 election campaign that simply looked like this basically highlighting his face that everybody knew with the mustache. And also hence creating our modern day association of this style of mustache with him and Chaplin.

Sources:

• Christopher Oldstone-Moore: Mustaches and Masculine Codes in Early Twnetieth-Century America, Journal of Social History, Vol. 45, No. 1 (Fall 2011), pp. 47-60.

• Ron Rosenbaum: The secret parts of fortune: three decades of intense investigations and edgy enthusiasms. 2000.

• Ludolf Herbst: Hitlers Charisma: Die Erfindung eines deutschen Messias 2010.

• Becoming Adolf in Vanity Fair.