I think there is an issue where the term 'genius' and 'prodigy' are used interchangeably in popular discourse, when they are in fact not the same.

Most 'child geniuses' are in-fact child prodigies, children which display extraordinary talent in one or multiple fields (say mathematics, piano, etc.).

Genius requires an element of creativity and not just proficiency. Schopenhauer made the distinction quite nicely: 'Talent hits a target no one else can hit; Genius hits a target no one else can see.'

I think very few people are both prodigies and geniuses. One notable example is Mozart - played the piano at an incredibly high level at a very young age and also composed some creative masterpieces later in life.