Dabi is new to your boyfriend, in front of others he pretends he doesn't know you for a simple reason: he wants to protect you. But for pretending he doesn't know you and being cold to you that makes up for it when you're alone he's a completely different person he's alone with you

Dabi was currently sitting on the couch with you and hugging you "I'm sorry baby I'm cold and I pretend I don't know you, I just don't want anyone to hurt you, you know who I am and I have a lot of enemies....."