That's not a JMSDF boat, is it? If not then there shouldn't be any obstacle to a future JS Akagi.
Naming conventions. The current ships like Kaga and Izumo are named after Providences (as were the battleships of old). Akagi was named after a mountain like the cruisers. The DDGs like Kongo are named following this pattern (like the older battlecruisers they were at first). Akagi was started as a battlecruiser before conversion to a aircraft carrier. Kaga was a battleship before conversion started.