In Shining Force (I and II), you follow a main hero as he travels through a journey. You start out with a small party, around 5ish, and you meet more characters that join your force as you progress. Each character has a unique identity and personality; there are no generic units. From what I saw from that video, the battle system is about identical. You move units on a grid and select an action from a menu. The action takes you to a "battle view" where there is an animation of whatever you do (swing your sword, cast a spell, use an item, etc.), as well as the result (how much damage and experience). You end up with a few dozen units total, and you can select maybe 12 at a time as the active "party" to go on the battle field.
You gain experience per action you perform, and you get a level up every 100 XP. Once you reach a certain level, you can promote your character to a new class (e.g., Mage -> Wizard), where their progression upon level up improves.
The Shining Force games were some of my favorites growing up. I liked the battle system a lot, and also the story and different characters and classes.