The Mussolini regime and Vichy France were never in friendly terms. After the armistice of June 1940, Italy treated France as a defeated foe and refused to establish a normal diplomatic relationship. All relations between the two countries had to go through a special commission, the CIAF (Commission italienne d'Armistice avec la France). Italian authorities were overtly contemptuous toward the French: Italian members of the CIAF were forbidden to shake hands with their French counterparts. The Italian press mocked the Vichy regime, noting that France, unlike "young" fascist Italy, was run by old men such as the 84-year old Pétain and the 73-year old Weygand. Collaboration between Italy and Vichy remained minimal and uncertain, with up and downs, and never on the same level as the one-sided but fruitful collaboration between Nazi Germany and Vichy. At one point, the Italians were willing to discuss a reorganization of the Italian and French territories, with France giving Nice and Monaco to Italy, as well as some of its African colonies. Occasionally, relations were more positive, but overall fascist Italy and Vichy France did not see eye to eye, and whatever happened between them was mostly under the influence of their relations with Germany anyway.

So, in November 1942 and following Operation Torch in Northern Africa, Italy invaded and occupied the South of France, obviously without asking for French permission. With the whole of France occupied either by the Germans or the Italians, Vichy had lost any reason to believe in its own sovereignty and protested in vain to the Italians. After the fall of Mussolini in July 1943, Italy, under Badoglio, formally switched over to the Allied. Then Mussolini was freed by the Germans and the Republic of Salo was created a few months later: nevertheless, Vichy France considered that its relationship with Italy was over, even with the puppet Mussolini regime. France's less-than-cordial agreements with Italy dating from June 1940 armistice were declared void. The CIAF was dissolved. Italian occupation of Southern France ended and was replaced by German occupation. While some German-controlled countries did recognize the Republic of Salo, the Germans did not pressure France into doing the same, only asking that Vichy maintained relations with Italian diplomats. Vichy had thus no particular reason to recognize a regime led by people that had always been hostile to them, and that was politically inexistent, being a German creation. Dealing directly with the Germans was sufficient.

Source: Rainero, Romain H., ‘La Collaboration Vue d’Italie’, Relations Internationales, 107, 2001, 419–28. https://www.jstor.org/stable/45345192