The dorky, silly side of the otherwise cool and collected Nagato. This frequently appears as a portrayal of Nagato as a "lover of cute things" (which can mean cats, squirrels, destroyer girls, Hoppo, the Azur Lane Nagato, etc.) to a pathological degree, even including outright pedophilia and kidnapping.
References to the "Nagamon" nickname are also acceptable.
Part of the reasoning is that she and Mutsu were heavily advertised towards Japanese children back in the 1920's as Japan's then-mightiest battleships, so Nagato the ship-girl is often joked about as a "friend to all children" (which she usually comically fails at being).
This joke is referenced in both the 4koma and episode 8 of the official anime.