I guess genes can have some influence on it, at least indirectly. But by far what seemed to be the biggest impact was the musical surrounding and transfer of experience. If you were taught by the best composers in the world, and if you happen to have a little bit of musical talent on top, it's just very likely you'll eventually become a very successful composer if you put some work into it.

Mozart was taught almost everything by his father, who was an excellent musician (composing maybe not so much) and initiated him with stuff like Bach. Mozart and Beethoven met many times and it's likely Mozart gave him some lessons as well. Beethoven was also a student of Haydn (another very big composer), then Beethoven was Czerny's teacher, which was in turn Liszt's teacher (maybe the most famous pianist in history)... The more you dig into it, the more links you find between the most renown musicians / composers.