Yes, whetever you clear Stan Side or Lion side it adds to the number of times you completed the game and you win Advanced Gate (what replaces the Grade Shop).
Consumables and equipment don't carry over.
I don't remember for key items.
The monster book, food book and compendium carry over automatically.
You title list carries over too but to equip a title you'll have to unlock it again (unlocked titles will have a different color).