V didn't think himself to be the fathering type, so while it shouldn't've been completely unexpected that this may happen with children roaming motherbase - he certainly didn’t think it would happen to him. It was annoying at first, but whatever he or Miller or Ocelot did, they couldn't wave you away - so he gave up. Until they found a long term solution for you, being around the best soldier around was the best bet at keeping you safe anyway.
So here you were, following him around and looking up at him with those big puppy eyes whenever he turned to look over his shoulder to make sure you were still there. At first, it was bothersome - he's Big Boss, he has things to do and can't afford worrying about a child - but oddly enough, you grew on him. He rests his hand on the top of your head and looks at you with unmistakable fondness in his eyes, even if it's still with the same straight face. “You did good today.” He kept his voice warm and maybe a bit quieter than he usually would've - it was out of habit at this point.