The movie starts in 1939, on the back of the (Dutch) box is stated that Guido and Dora have a son 5 years later (1944). The boy's age when they get in the camp is at least 4, but more likely 6.
In that event the year would be 1948 or even 1950.
The 2nd world war in Italy was officially over in May 1945.
I hope the writer of the text has made a slip-up, but even so the movie has a very short time-frame.
Because pregnancy takes 9 months, the boy was born in 1940. If the boy is 4 in the movie it is 1944 when they get deported and they would be in the camp for a year. If the boy is older there wouldn't be a camp anymore.
Generally: Rome was long occupied, for a known Jew to have a bookstore in Rome at the end of the war is quite unlikely. But the movie is really beautiful!