Remus had always loved dogs, even before he met Sirius. Being a werewolf, he didn't trust himself to own one, so he never had his dream pet.
Once he met the other marauders, he of course figured out immediately that Sirius was named for the dog star. So he wasn't exactly surprised when Padfoot's animagus was a black dog.
James never turned into his animagus unless it was a full moon night, and Peter used his only for pranks when he needed to get into small places. But Sirius was always turning into a dog, just so he could lay in Remus's lap and have the boy scratch behind Sirius's ears.
It was especially useful when trying to convince Remus to pay attention to him instead if homework, because he had for-real puppy dog eyes. Sirius knew Remus couldn't say no to that sweet face.