Candy's relationship to his dog can be compared to George's relationship with Lennie. What are some similarities between the two relationships,...

George and Lennie have been buddies for a long time. George has gotten attached to Lennie just as Candy is attached to his old dog. George feels a sense of responsibility for Lennie, just as Candy feels for the dog. Lennie could not survive without George, and the old dog could not survive without Candy. George is emotionally moved by Lennie's trust and devotion. The same applies to Candy's feelings about his dog. 

There is...

George and Lennie have been buddies for a long time. George has gotten attached to Lennie just as Candy is attached to his old dog. George feels a sense of responsibility for Lennie, just as Candy feels for the dog. Lennie could not survive without George, and the old dog could not survive without Candy. George is emotionally moved by Lennie's trust and devotion. The same applies to Candy's feelings about his dog. 

There is definitely a strong implicit analogy between George's relationship with Lennie, on the one hand, and Candy's relationship with his old dog, on the other. However, Steinbeck had another reason for inserting the episode in which Carlson pressures Candy into letting him shoot his dog. George is going to shoot Lennie in the last chapter. He will have to have a gun. He can't risk using any other kind of weapon because Lennie is much bigger and stronger than he is. He might end up getting killed himself. When Carlson shoots Candy's dog he reveals where he keeps his German Luger. After the shooting, Carlson carefully cleans the Luger and puts it away. George is able to observe how this complicated German automatic pistol works. The bullets are contained in a clip that fits inside the handle. The first bullet is jacked into firing position by pulling back on the ejecting mechanism.  

Steinbeck called his book "a playable novel." He had an arrangement to adapt the book into a play immediately, which explains why all the exposition is contained in the dialogue. Both book and play came out in 1937. Steinbeck wanted the theater audience to realize when George pulled the distinctive-looking Luger out of his coat-pocket that he had stolen it from under Carlson's bunk. All the men in the mob thought Lennie had stolen it, and the audience did not know otherwise until they see it in George's hand. Then they understood that George had intended to kill his friend ever since he saw the girl's dead body in the barn. Lennie was getting to be a menace, a potential serial killer of young girls. He had to go.

So the scenes in which Carlson kills the old dog and then cleans his Luger were primarily intended to show the gun and show George how to shoot it. Carlson even explains how he will kill the dog painlessly with one shot by firing into the back of its head. So George knows how to kill his friend painlessly by shooting him in the same place. This also is the conclusion of the analogy with Carlson and his old dog.