The most simple answer is that the political system did not change.

It's a fact that is often overlooked and an extremely complicated situation but I'll try to summarise.

At the political election of 1921, the Partito Nazionale Fascista only gained 35 MPs, making it the sixth 'largest' party in parliament. And I say 'largest' as it was the sixth party out of 15 who had MPs elected in parliament. Therefore, as expected, it did not have the faintest chance to get anywhere near government.

The March on Rome in 1922 was not a coup d'etat in the traditional sense where the Fascist Party violently took control of institutions and unilaterally declared itself the governing party in Italy. It was a show of strength aimed at forcing the King to give Mussolini a mandate to form a government, and that is exactly what happened. The King simply dismissed the sitting prime minister, liberal Facta, and give Mussolini a mandate to form a government.

The oddity, and major difference with Nazi Germany, is that Mussolini never actually suspended or revoked the constitution, the King remained the nominal head of state and technically speaking Mussolini's mandate could be revoked at any time. What the regime did was simply pass laws within the existing constitutional framework with many of those laws and reforms actually surviving the fall of fascism and the republican constitution.

And that is a key point, the King could have revoked Mussolini's mandate as prime minister at any time. From a practical point of view this was not possible until 1943 but that is exactly what happened after 25th July 1943. The Fascist Grand Council passed a resolution calling for Mussolini's resignation and the King took this as the green light to dismiss Mussolini as prime minister and replace him with Marshall Badoglio.

And that's another often overlooked fact, Mussolini was not deposed through any revolutionary act or through violence but was much less glamorously dismissed by the King after his party demanded his resignation. The fall of fascism as government of a unified Italy (because the Repubblica Sociale Italiana is a completely different beast), happened within the Italian constitutional framework of the time.

That's why the flag was not changed until the proclamation of the republic, because Mussolini was a prime minister of the Kingdom of Italy.