“And I suppose this gets at the Big Idea behind Dogmeat’s relationship to you in Fallout 4. Like your character, Dogmeat is caught out of time. This dog doesn’t belong here, and neither do you.*”
The Dog in Fallout Demo is kicked out of the house by his owner. The parallels are obvious there.
Dogmeat in Fallout 1 lost his owner, who was a reference to Mad Max 2, and it’s pretty obvious that the early games especially draw a lot from that series. He is begging someone to open the door, let him in, let him have a home, and he will never succeed. His only recourse is to move on with someone else. You can win him over with food, and you are on a quest for water. No matter what, he is canonically doomed by the narrative.
There are several dogs in Fallout 2. K-9 and Robodog are both literally shaped by the Enclave like you end up figuratively shaped by them. K-9 specifically was forced to watch as the bastard Enclave scientist he tried to kill conducted horrible experiments, not unlike how you are helpless to stop the devastation of your people or the extermination of the talking deathclaws. Dogmeat is the same one from Fallout 1, a special encounter found in a time traveling pop culture reference/meta moment—because this is Fallout 2, and anything goes, no matter how wacky. You receive him by showing your vault jumpsuit to your ancestor—proof that you are the Vault Dweller’s legacy, that you are stuck dealing with the Vault Dweller’s legacy, that even in this extradimensional location beyond time and space you cannot leave behind the Vault Dweller’s legacy any more than Dogmeat can leave behind his first owner (who attacks if you hurt Dogmeat). Finally, Pariah Dog is an unlucky but very hard to kill wretch who brings ruin wherever he goes.
Fallout 3’s dog, as Anon mentioned, is lost and fighting for his life with his owner gone–maybe you should have been worried about the fact that the James analogue here is already dead. He can come back to life through the power of DLC by producing puppies–a promise of new life after devastation.
Fallout: New Vegas has Rex, who was pulled apart and put back together just like you. You survived brain damage thanks to a kindly stranger’s intervention, and you can pass on the favor by giving Rex a new brain to replace his aging one. In a game deeply preoccupied with metaphorical kings, he is literally named “king.” He has an extensive backstory of traveling around the Wasteland, but most of it isn’t really relevant right now. He is cybernetically enhanced, and if you play Old World Blues, you will be too (one way or another). Also, he likes The Kings.
Fallout 4’s Dogmeat is a perfect German Shepherd in a world of mutants. It is a dog out of time, waiting for the chance to be with a loved one. It is the final piece of your white picket fence dream, 200 years too late.