Fast seemed overcome with sadness, and it took me a moment to understand: Sister, his favorite companion, was gone, as lost to us as was Hungry. (1.59)
The dog isn't particularly loyal to his own doggy family, but that appears to be a trait unique to him, for whatever reason. Fast and Sister are loyal to one another…although, okay, Mother is only loyal to herself.
I crouched low and licked them all in the face, letting them know in no uncertain terms that they'd have zero problems with me—it was my brother who was the troublemaker. (2.33)
Our narrator seems like a bit of a coward here. He casts his lot with the big dogs in the pen, and he doesn't stand up for his brother or his family. On the other hand, he is just a puppy.
I was torn between loyalty to Mother, who had fed me and taught me and taken care of me, and to the pack, which included my worthless brother, Fast. (2.46)
"Torn" seems to be a strong word here, as our narrator has never expressed any sort of loyalty to his mother or to his siblings. Are you surprised when he leaves with her? (On the other hand, it's not like she exactly sticks around after a certain point or anything.)
Spike was unquestionably the leader now, a message he enforced by challenging every single one of us, head-to-head in the yard. (4.34)
Spike is seen as a bully, a villain of sorts among dogs. But how is his technique to gain superiority different from Jakob's? Jakob, of course, eventually finds a dog that will submit to his control, but the dog loves Jakob in a way s/he didn't care for Spike.
I put on an extra burst of speed and leaped through the air, landing on the back of his go-kart and nearly toppling off again. (12.36)
This is a "Race for Your Life, Charlie Brown" moment, when Bailey helps Ethan win a go-kart race. Except Bailey is much more helpful to his owner than Snoopy ever was.
"None of them like it. The question is will she stop struggling and let me be the boss, or will she keep fighting? I got to have a dog that knows I'm the boss." (18.6)
Jakob isn't looking for a cuddle puppy: he wants a dog who is loyal and obedient. Ellie, as the dog is now known, can definitely be loyal—although we'd be lying if we said she didn't miss the snuggles a bit.
It was, I reflected, as close as Jakob could come to the unrestrained adoration I once felt from Ethan." (19.25)
When Jakob says "good dog" to Ellie, it means something different from what "good dog" meant when Ethan said it to Bailey (back when Ellie was Bailey). Jakob and Ellie have a working relationship, one based on professional bonds of loyalty instead of gooey notions of love.
I lay there quietly with my boy in the stillness of that spring afternoon, the house silent and empty. Soon the girl would be home, and, remembering how hard it had been for everyone to say good-bye to Bailey and Ellie and even the cats, I knew she would need my help to face life without the boy. As for me: I loyally remained right where I was, remembering the first time I had ever seen the boy and then just now, the very last time—and all the times in between. (32.47)
Here we see Buddy actually use the word "loyal." Loyalty means showing constant support, no matter what, and Buddy remains loyal to Ethan, his "boy," even when Ethan is an old man. Buddy realizes that loyalty extends even beyond Ethan's life: he must support Ethan's family, too. He knows Ethan would want that.