It's called Flower Memory and it replace the old promotion abilities (the things that add speed, turn 1 ATK, HP limit break, etc). Now it works a bit like equipement but you attach it to a place in the team instead of equiping it to a girl. You can get them through gacha with Memory Seeds (looks like rainbows seeds) , you can level up them with specilal manyu, and you can fuse dupe to increase level limit and power up the memory effect. And it give bonus HP that increase with level.
The old promotion abilities still exist as bronze Memory and new ones are silver, gold or rainbow. Most of them are meh, some of them are very good.

Glad to see you arround here again. I hope you will stay a bit and enjoy