Mussolini came to power in what amounted to a coup d’etat (though not a military take over in the sense of the armed forces seizing control of the government; Mussolini’s paramilitary Fascist organization seized power). This seizure was later ratified by way of rigged elections; however, many Italians genuinely welcomed Mussolini’s rise to power. There is a common saying “Well, at least he made the trains run on time”–not necessarily true, but there is generally a perception about leaders of this type that they “restore order” in the face of political chaos (middle and upper class fear of working class socialist or communist movements) and economic dislocations (the Great Depression). There are frequently nationalistic grievances as well–Italy was on the winning side of World War I, but many Italians felt it hadn’t gotten its just desserts in the way of territorial gains; Germany was on the losing side, and there was much popular German resentment of what were seen as the unduly harsh and punitive terms of the Versailles peace treaty (territorial losses and allegedly crippling reparations payments, plus severe limits on the size and strength of the German military).
Hitler came to power by initially democratic means–the Nazis won a plurality, though not a majority, in parliamentary elections; the Nazis (initially in coalition with some other parties) formed a government in accordance with the parliamentary rules, but soon consolidated dictatorial power by measures rammed through the Reichstag (parliament), extralegal means, and popular plebiscites. Like Mussolini, Hitler’s rule was initially generally quite well-received by many Germans.
The leader of a fascist country had essentially unlimited power–exaltation of the single supreme national leader was a hallmark of both Italian Fascism and German Nazism, and generally of other similar regimes in other countries. Fascist states don’t have provisions for impeachment proceedings–about the only way to get rid of a fascist leader is to kill him. Like “absolute monarchs” of the past, in reality fascist leaders did have to take into account the interests of other powerful groups in society, particularly the armed forces, which were generally power bases which predated the rise of fascist leaders, but were usually co-opted into supporting the new regimes. (Military re-armament and foreign conquest and the general glorification of all things military being common policies in fascist states.)
Generally speaking, other countries recognized the fascist leaderships, just as we recognize assorted dictators and unelected governments today. Technically, Mussolini’s Italy was still a monarchy, with the king acting as a powerless figurehead, as in other constitutional monarchies, only for an arbitrary dictatorship rather than a parliamentary democracy. (Semi-fascist Spain under Franco in theory had a somewhat similar set-up.) In Germany, there was initially a parliamentary republic style President (i.e., one without direct power over the day-to-day government of the republic, though more than a typical modern constitutional monarch enjoys); when Hitler was first made Chancellor (prime minister) Paul von Hindenburg, a retired military officer, still held the office of President; however, when Hindenburg died, Hitler consolidated the powers of the Presidency and the Chancellorship into a single office of Fuehrer (Leader) and Reich Chancellor, and thereafter held supreme personal power over the German state.