Can't say I played much of Kamihime to compare them.

Parties are 5 girls +1 extra and 1 helper which switch in for the dead.
Battles are sorta real-time with turns. Selected skills are activated immediately after whatever current animation finishes and can be chained for a damage bonus. Skills have a recharge time but it only fills a set amount per action.

Each girl has one fixed personal skill and can equip up to three subskills of certain categories/colors. Their equipped weapon can provide a fifth skill.
Orange = physical attacks
Blue = auxiliary attacks (added status effects/lifedrain/etc.)
Purple = magic attacks
Pink = support magic
Green = passives

Unwanted girls can be scrapped for a copy of their base personal skill as a subskill.
Subskills are treated like items that can be equipped and unequipped by anyone that can use them.
Skills are leveled up with generic skill fairies or copies of the base skill. Max skill level is 5.

Rainbows & golds have their personal skill evolve with their second limit break.
Limit breaking girls costs a bunch of farmable elemental stones or a single dupe. Using dupes is not recommended outside of maybe the last limit break for rainbows/golds. Otogi Stones can be in short supply.