Is it true that Mussolini required all family names of non-Italian origin to be italianised?
Yes, there was an active policy of Italianization supported by the fascist government. In Parliament, specifically the senate, legislative activity was mostly headed by Ettore Tolomei.
There was no unified policy or body of law favoring Italianization. Rather, the senate commissions on education, foreign affairs, and integration of executive decrees (all of which Tolomei was a member) set forth a number of initiatives favoring Italianization; notably in 1928 a law was passed making it illegal for priests (keepers of baptismal records) and census taker to register "foreign" surnames. Notaries and census-takers were instructed to transcribe names in more acceptable "Italianized" form (sometimes without the consent or knowledge of people whose names were being changed) disproportionately impacting residents of border regions like the Giulia (specifically, the Carnaiola), a region in the far northeast of Italy near Slovenia in which the large city of Trieste is located. Names like Maldič were italianized as Maldini, Rusovič as Russo and so forth. In the city of Trieste, the "translation" of surnames was mired and systematic. However, there were notable gaps, for example in the more cosmopolitan Milan, Giulian surnames like Gaberščik slipped through the cracks.
But Italianization didn't stop with surnames. Already in 1923 a law was passed decrying that place names were to be Italianized, and signage bearing foreign-sounding place names like Courmayeur were substituted with the more "Italian" Cormaiore. Lastly, dictionaries and school curricula removed foreign words; "Sandwich" for example, became "Tramezzino." Some "Italianized" words stuck, others, like "Bevanda Arlecchina" in the place of "Cocktail" immediately became passè. Business leaders and leading fascist intellectuals, like Giovanni Gentile, the publisher of a major encyclopedia, encouraged the Italianization initiatives in the private sectors (there were some holdouts, notably Sports kept using lots of foreign words). There soon came to be a great stigma attached to having a foreign-sounding surnames which could impact people's careers, especially in the public sector.
There were also mired policies to populate foreign-speaking parts of Italy with native speakers of Italian, with incentives for native speakers of Italian to invest in and open businesses in Italian-minority locales. Notably, the city of Bolzano went from being majority German-speaking to majority Italian-speaking.