Sorry it isn't very clear in the demo, we've improved and explained it better in newer versions. Gifts can only be given once per day, so if it seems like she's not accepting the gift, it's because you've already given her one that day. Wait for the next day and try again and it SHOULD trigger the gift giving sequence.
Hmmm that might require a save restart if you're playing on 0.01B. I believe we've implemented the gifting tutorial at the beginning, which teaches the player how to gift items before unlocking the ability to sleep and or use the dating (SMS) app. If that gifting thing isn't working, maybe a trigger was missed. Let me know how that goes? Another requirement is to add cards into your deck using the deck manager and also opening the wardrobe at least once.