Does the game have some sort of 'pity system'?

Yes, the game has a similar system that will allow players to get a guaranteed ranked unit after every X number of attempts. Furthermore, the counter towards the pity will be retained under the same banner. So if ever you fail to trigger the pity from one banner, the next banner type will allow you to continue where you left off. Of course, this pity counter will reset after getting a guaranteed S-rank character.