Yes, they do, but it really ranges. You will see plenty of kids wearing Disney themed clothing onboard, especially shirts, dresses, and Marvel gear. There are generally fewer Mickey ears than you see in the parks, but Disney style outfits are still very common.
During the day, most kids are in swimsuits, cover ups, or casual clothes as they move between the pools, kids clubs, and activities. Comfort tends to win out during sea days and port days.
Dinner attire varies widely. Some kids show up in full princess dresses or character outfits, while others wear simple dresses, shorts, or polos. Disney Cruise Line is very relaxed when it comes to what kids wear in the main dining rooms. The general rule is to wear what feels comfortable and appropriate for your family.
One reason you see more dressed up kids on certain days is Bibbidi Bobbidi Boutique. This onboard experience offers princess and knight makeovers that many families love. Kids who participate often stay fully dressed up for the rest of the day and into dinner, which adds to the fairytale feel of the cruise.
If your child loves costumes, bringing outfits from home is always an option and can save money. If they prefer to stay casual, that is perfectly normal too. On a Disney cruise, there is no pressure either way, and kids fit right in whether they are dressed as royalty or ready for the pool.