James, 

the simplest answer is because each child has only 1 back, and in French it doesn’t make sense to say ‘ les dos de leurs amis ‘ as this would indicate each friend had more than 1 back. It could however be, for example, ‘ les bras ou les jambes de leurs amis ‘.

The sentence in French conveys that 1. each child has more than 1 fish 2. each child has more than 1 friend 3. each friend has only 1 back

Remember French focusses on the number of possessions to determine plural or singular, with 3rd person plural possessions being  ‘tricky’ and even ‘really tricky’ as Laura Lawless describes in the attached. Although she does not cover this exact situation, the basic principle is the same - using the plural for the possession in 3rd person indicates each person having multiple of the ‘possessions’. 

https://www.lawlessfrench.com/grammar/possessive-adjectives-plural-possessions/

Thanks.  Got it.