Disclaimer: None of this belongs to me, it all belongs to JK Rowling.
Lyra couldn't remember being loved. She had witnessed it for many years, been within a family and observed love between her aunt, uncle and cousin, but she herself had been ostracised from that love. Made to feel she didn't deserve it. She had also been prevented from ever getting close to other families and seeing their love as well.
She had, to an extent, felt it when she had stayed with the Weasleys the summer before her second year. But then she had had to acclimatise to being in close contact with people that she had never met before and living with them for several weeks. Although she had felt welcomed, she still felt slightly out of place. After all, she hadn't known Mr and Mrs Weasley, or Ginny, and suddenly she was their guest, in a house she didn't know.
So the first time she had got a letter from Sirius once she was back at Privet Drive for the summer, merely a week since she had met him, and read the words 'I love you', she had sat and cried. Yet, somehow, this felt right. Lyra felt loved, for the first time in her life. So she didn't have a problem, when she replied, to write 'I love you too'.
It was the sight of those words, written in his goddaughters neat script, that sent Sirius in to tears. He hadn't cried since the day he handed Lyra over to Hagrid, nearly 13 years before. He hadn't let the tears overcome him in Azkaban, he hadn't cried the day he saw Lyra the summer before when she escaped Privet Drive.
But this...
Hands shaking, Sirius sat next to Buckbeak and kissed the letter. He knew, more than anyone else, how Lyra felt growing up. He had felt the same, in his 'childhood' at Grimmauld Place, and knew what it meant that she had said that to him. He knew that he would do anything to protect her, he knew that they were a family.