A few points that rule out ship age as a deciding factor:
- The Nagato and the Mutsu are the secretary ships of the fleet and hold highest seniority, yet the construction of the four Kongou-class battleships had all been completed at least five years beforehand. The Kongou-class were all originally battle cruisers, which is a possible reason for their lower seniority assuming it's based on ship class, as the Nagato-class were dreadnoughts, and the Nagato often served as flagship IRL.
- The Ashigara and the other three Myoukou-class heavy cruisers are depicted as teachers. However, every single Mutsuki-class destroyer was completed around 4-5 years before them, and they're Ashigara's students in the show.
- The Ooyodo light cruiser was completed in 1943, well after most other ships in the fleet, but works with Nagato and Mutsu, which is likely because she was originally intended as a command vessel for submarine operations and briefly served as flagship of the Combined Fleet before being sunk.
- The Shimakaze was also older than the Mutsu.
Given the other references to real-life attributes of the various ships and their histories (e.g. the Sendai fought all of her battles at night, hence her character's obsession therewith), I would guess fleet seniority is handled based on ship roles in the actual Combined Fleet, which would roughly follow ship class just because of the different capabilities/stats of each.
The information of the ships are referenced from Wikipedia