Loved refers to that past, since it's in past tense.
Or often it refers to an idea that we all loved, but didn't happen/work. Actually, usually that's what it means.
This case it's a bit of a broad sentence, it could be either that the idea of "a puppy and a girl" growing together didn't happen, but they really loved that if it happened. Or it could be that the puppy and the girl already are older, and they were mentioning the good experiences from the past of watching them grow together.
The above are the two most probable possibilities of what this sentence could mean.
To be honest, this sentence isn't super grammatical... There are definitely different ways to express his/her sentence.