Since Naruto games often have "Story Mode" and "Versus Mode" separate, the type of save file matters:
Pro Tip: If you download a 100% save file, make sure the region matches your ROM!
A US save file will not work on a European ROM and vice versa.
Here is the actionable guide. Follow these steps carefully.
Cause: Your memory card is full, or the save is in the wrong slot (Slot 1 vs Slot 2).
Fix: In Aether SX2 settings, go to General > Memory Cards. Ensure the card containing the save is set to Slot 1 (Port 1). Most PS2 games only read Slot 1 on boot.