First comment on this sub, so I'll give this as much of a shot as I can. Here I'll be answering in the context of Nazi Germany and Fascist Italy, (since, as a previous commenter pointed out, the origin of these points was postwar Italy) but other fascist regimes from Pinochet's Chile to Idi Amin's Uganda follow these rules and break them as well, to varying extents.

The best way to tackle this is to go down the list and discuss its accuracy or fallacy:

1. Powerful and continuing expressions of nationalism: Italy and Germany were undoubtedly nationalistic and committed to their ideal of a New Italy or a New Germany, respectively. In Germany, the 1936 Summer Olympics were engineered specifically to show off the new Germany to the rest of the world, in all of its athletic and racial glory. The games were one way for nationalistic Germans to prove their nation's superiority, and was a great propaganda event as well -- Hitler commissioned a propaganda film to be made called Olympia, which was produced by the same filmmaker who made another famous Nazi film called Triumph of the Will. As for Italy's nationalism, I will pull a quote from The Doctrine of Fascism (1935) because the book relevant in both this point and a few others: "The Fascist conception of the State is all embracing; outside of it no human or spiritual values can exist, much less have value. Thus understood, Fascism, is totalitarian...No individuals or groups [will be allowed] outside the State"

2. Disdain for the importance of human rights: There is a wealth of well known examples to pull on here for Nazi Germany (namely but not limited to the Holocaust), so I'll comment on Italy, which is perhaps lesser known in this situation because Italy largely did not participate in the mass murder of Jews until being taken over in Germany in 1943 (although antisemitism still ran rampant). There, the blackshirt militia was commonly used to violently put down strikes, riots, and destroy political opposition (which would later be used as inspiration for Germany's SS), resulting in the murder and maiming of thousands, the torching of newspaper presses, and the silencing of speech.

3. Identification of enemies/scapegoats as a unifying cause: Jews can be pointed to in both countries, moreso in Germany of course, where the NSDAP championed the 'Stab in the back' myth, which essentially claimed that the Great War was not lost by the German soldiers themselves, but scheming saboteurs that impeded the war effort at home (namely Jews -- it didn't help that the Treaty of Versailles' German signature was made by a Jewish diplomat). In Italy, hate was often directed at the Entente of the First World War. Even though they fought on the same side, nationalist in Italy felt that the Entente powers Britain and France, who had the largest part in the writing of Versailles, cheated them out of territory that was rightfully theirs, namely Dalmatia, which was in Austria-Hungary. This lead to the propagation of the idea among nationalists of a 'mutilated victory' in which Italy won the war but was betrayed or forgotten by their former allies, and that the Italian people had to compensate for this stolen territory by invading other countries and retaking it -- in this situation, the scapegoat is the Entente powers, even despite the territorial gains Italy enjoyed such as Tyrol, Istria, and a few Aegean islands.

4. The Supremacy of the Military/avid militarism: As just mentioned, Italy's militarism was manifested through the Blackshirt raids on political enemies and through the mutilated victory myth perpetuated by pseudohistorians. Also unrelated (mostly) to World War 2 is the Second Italo-Ethiopian war, which was a display of militarism and a sort of revenge for the first war (which was a humiliating Italian defeat). For Germany, expansionism was always a goal of the Nazi government, and among many displays of military might, the remilitarization of the Rhineland is perhaps the greatest, because it directly and obviously breaks both Versailles and the Locarno Treaties (written after the Entente withdrew from the Ruhr). Not only did they enter with troops where they were not legally supposed to, but they made a celebration out of it, as somewhat of a proclamation to the world of Germany's direction and strength.

5. Rampant Sexism: Up until here, the points have been concrete and easy to prove with a large array of evidence, but this is the part where things get dubious. In Italy, women's role in the wartime economy was similar to that of the United States: they were seen as housemakers for their husbands, and when they were fighting, the women would serve as factory workers to fuel the nation's home front. Although the idea of a 'model homemaker wife' is sexist by today's standards, it doesn't seem to be out of the norm for a country that is both in the 1930s and largely Catholic. Nazi Germany was notably more sexist and also focused on the motherhood aspect of women. Given the time period it's not entirely out of place, but it should be said Weimar women did enjoy much more autonomy and political freedom under the previous Weimar Republic than under the Reich. Whether each regime's views on women were 'rampantly sexist' or not is up to reader interpretation.

6. A controlled mass media: Undoubtedly yes. As previously mentioned, Mussolini was quick to suppress free speech and internal political opposition where he could, while Hitler was also very strict with not just the press, but also cinema and music. One of his top and most famous cabinet members, Joseph Goebbels, is remembered for his time as the "Reich Minister of Public Enlightenment and Propaganda" While he is responsible for a number of propaganda films, (such as previously mentioned Triumph of the Will) I think his name tells you all you need to know.

7. Obsession with National Security: If I were writing my own list of fascist warning signs, I'd probably nix this point because it redundantly falls mostly under the category of militarism and nationalism, and how much emphasis counts as 'obsession' is somewhat subjective, but for the record both Italy and Germany's fascist government invested heavily in destroying opponents who could pose a threat to national security, such as the famous Night of the Long Knives in Nazi Germany, where former Nazi associates and government officials, and really anyone who could pose a threat to the new government, was murdered.

The museum you're referencing (where you saw the poster) is the United States Holocaust Memorial Museum, one of the preeminent museums and research facilities about the Shoah, Holocausts and genocides in general, and white supremacy.

https://www.ushmm.org/