The mysterious little girl fell from the skies and made her entrance with a kiss. No one knew exactly where she had come from or who she really was, and she certainly wasn't about to tell anyone. Many accuse Chibiusa of being a mean-spirited brat, but the situation needs to be looked at from a different angle. Chibiusa enters present-day Tokyo alone, young, and carrying the burden and guilt of the devastation she left behind.
Mamoru seems to be the only person Chibiusa really quickly warms up to when she first arrives. She notes that his embrace is warm and familiar like her father's. He feels a connection with her through his psychometry. She trusts him, and clings to him almost instantly.
Her bond with him deepens over time, especially once they know her true identity as Usagi and Mamoru's future daughter. He trusts in her and wants to protect her just like Usagi Usagi and Mamoru play more the part of older sister and brother to Chibiusa in present-day Tokyo, but both can be very parental at times.
As Chibiusa spends more and more time there, Mamoru watches her mature and accomplish new things. In this manner he has a very fatherly kind of pride. Next to Usagi and Mamoru's bond, Mamoru's relationship with his daughter is one of my favorite things to watch develop and grow over the course of the series. Mamoru seems to just be innately good with children, extremely loving and at ease with them as well. I think that when Chibiusa first arrived, he really helped her feel more at ease; he gave her some feeling of protection and warmth in this world where she thought she was utterly alone. Once Chibiusa realizes that Mamoru is her future father, I believe she begins to want and appreciate the attention he gives her even more. Perhaps in the future the King and Queen can't be so laid-back with their princess, so she makes the best of her leisure time with them in the 20th century.