Great/interesting film. Ferrari’s wife, as part of their financial arrangement, insisted his biological son never use the Ferrari surname. The film didn’t specifically say he agreed or disagreed. However, during the film’s credits, it referenced his son, Piero Ferrari, having entered the business.
So, apparently his son did use the name. But did he do so only after Enzo’s wife passed?
Yes. Laura Ferrari died in 1978. Piero Ferrari legally changed his name in 1990.
How could she control his son’s use of the name as an adult?
The movie takes a bit of license with this.
Italy was (and in many ways still is) a very socially conservative country. Divorce wasn’t legal until 1971, and if Laura and Enzo had gotten a divorce at that point it might have been possible then for Piero to have petitioned to legally change his name. As it was, they did not divorce and so 1978 would have been the earliest year it would have been possible to even start to do so. I find it interesting that it didn’t happen until 1990 - two years after his father’s death.
I’ll add that Italy also takes names pretty seriously - I remember seeing an interview with the Italian actress Asia Argento years ago, and the interviewer asked about her unusual first name. She said that when her parents went to register her name with the local authorities in Rome, they refused to register Asia - she’s actually named Aria on her birth certificate. There was a list of acceptable names, and that was the closest one.