Basso continuo is just chord symbols written the baroque way. It isn't really improvisation in the same sense as Jazz improvisation.

You take your bass note and you recognize which inversion you are in by the figure under it and you play the chord.

Once you know your figures it is just a matter of familiarity.

Mozart had a talent for remembering what people had played. The film is a dramatization.

In baroque scores where "improvisation" breaks were included in the score, the "improvised" parts tended to be prewritten and were not spontaneous inventions as is the tradition in Jazz.