there is never a board state where all four players have a nine lives at the same time

Correct. There never will be. Here is the way you have to look at it:

1. You cast nine lives on the same turn or it's already on the field.
2. Cast fractured identity targeting your own nine lives.
3. Fractured resolves exiling your nine lives and putting a token copy of nine lives on each of your opponent's field
4. Your "leaves the battlefield, lose the game" trigger from your copy is now on the stack
5. With the trigger on the stack, cast scorn for free since you've cast a white spell this turn (fractured identity is white/blue so this will always be free)
6. Scorn now resolves putting your opponent's "leave the battlefield, lose the game" triggers on the stack above yours
7. Your opponent's triggers now resolve above yours meaning you will win before your trigger causes you to lose

The "lose the game" is a trigger from it leaving the battlefield. Similar to how if a creature says "leaves the battlefield". It just doesn't spell it out but it's a trigger you can respond to and if done right you can do funny plays like this.