You will receive a random date gift from the Sky Garden (directly deposited into your inventory as an unopened present) whenever you provide each girl with a lover's gift obtained through the Purchase Gifts function. Each respective lover's gifts can be identified by the pink heart in the right hand portion of the box for any given gift.
The inventory storage has 30 storage spots along with the 6 separate active date gift spots that you carry with you into each of the dates (puzzle matching sessions) you have with any of the girls. As there are other items you can accumulate in your inventory, other than the date gifts themselves, there will not be enough storage spots available to store all of the date gifts than can be gifted to the player character. This being the case, you will need to discard some of your less preferred date gifts.
A list of these items and what each of their effects is may be found below.