The in-game world in Days Gone is grim and dangerous, yet it does not mean that all creatures you encounter want to kill Deacon. This section of the guide includes the information on whether Deacon can pet the dog and befriend any animal.
Yes, in Days Gone you can pet the dog, however, this becomes available only during a cutscene initiated after you make sufficient progress in the story campaign. You have to complete main missions until you start the Have It Your Way mission. During this mission, Deacon is looking for a pup for Boozer.
In your search for the pup, you have to search through the nearby houses with freakers inside. The doggy is hiding in the privy and will not come out of it even after you kill the freaker trying to get him.
You can encourage the dog to come out from hiding by offering it some dog food. You have to start a mini-investigation. Search the blue circles on your minimap and follow tracks. You will find out that you have to climb the lower roof of the building and enter the room on the upper floor.
Find the Dog Treats in the bedroom and take it to the privy. Stand exactly in front of the locked door and hold the interaction button.
A cutscene will start, during which the pup will come out of its hiding and befriend Deacon.
Upon returning to Iron Mike's camp you will see another cutscene, this time showing Boozer adopting the puppy.
Unfortunately, not all scenes involving dogs are positive. During one of the earlier missions called SPOILER, Deacon is forced to put a dying dog to sleep. Deacon and Boozer get a chance to take the revenge to the murderers of defenseless animals.
When visiting Iron Mike's camp for the next time, you will find the pup in front of the dog house next to Bozzer trailer. The game, however, does not allow you to interact with the dog.
Later in the game, Deacon reaches the Crater Lake region. This is a great opportunity to get a toy for the dog. It is necessary to examine the hut in the mountains from the above image - it is located to the north-east of the Diamond Lake camp.
As a reminder, the hut we're talking about is the first visited location after arriving in Crater Lake. Deacon had to defeat a Rager (mutated bear). You may return to the house during one of the subsequent story missions (Prove It to Me) or you can visit it on your own initiative.
Explore the interior of the hut. Yellow Dog Toy is lying on the floor.
You may give the toy to Boozer's dog in the final part of the main plot or after its completion. After deacon returns to the northern regions of the world map, go to Iron Mike's camp and visit Boozer in his hut. Go to the dog and hold the interaction button to give him the toy.