In my personal opinion, a Fallout game is a game that takes place in the Fallout World, continuing the setting and themes of the previous games, which have generally been commentary on humanity, mainly its self-destructive nature in pursuit of short term ends (Resource Wars leading to the Great War), its intrinsic tribalism (us versus them mentality), and, despite all of that, its power to keep moving forward. A Fallout game does not have just that, but offers the player a variety of role playing elements to create themselves or a sort of wasteland alter-ego to explore this world and decide how they are going to go about learning and not learning these listens (maybe even one right after the other.
Now I am going to go through each game to say what I think about all of these factors.
Fallout 1 did a good job overall. It had some limitations that kept it from being the perfect model of this, such as how everybody besides the Unity was painted as generally good while the Unity was painted as generally bad, using cults and being hostile to you and everybody who has ever been friendly to you, and the fact that no matter what you grew up in a Vault, but that hardly mattered. Once you were in the Wasteland, the choices were flexible and the endless examples of self-destruction, tribalism, and progress were everywhere. My favorite example of its is Vault 15, where the four factions that emerged from it (Khans, Jackals, Scorpions, and Shady Sands) showed a variety of traits in those three categories (even though we really only get to know the Khans and Shady Sands in it, which were appropriate choices to pick consider they showed the opposite ends of the spectrum).
Fallout 2 was even better with factions but still gave a pretty focused background. The Enclave were about as reasonable as the Unity but those who were on your side were generally darker as well. The Brotherhood of Steel's ideology had reached its extreme and now instead of just seeming like a bunch of overall good military men, they came off as more xenophobic and secretive. Despite the fact that New California was being united under the New California Republic, they didn't really come across as absolute good guys. The fact that the NCR was modeled after the Old World US greatly contributed to the idea of mistakes being repeated.
Fallout Tactics had much less RPG elements but a great story to tell despite, or perhaps because of it. This is why I am generally ok with calling Tactics a sort of Fallout game. I enjoyed the Reavers and the MWBoS and Barnaky and how the entire region really comes across as broken by how far back so many have fallen. It's dominated by tribes, violent raider gangs, religions that worship the pre-war world, and an army of mysterious robots before the extremely militant MWBoS jumps in to make things even more complicated. I always appreciated how those who were considered the progressives of the Brotherhood of Steel weren't what we would consider modern day progressives (humanitarian, egalitarian types) but were violent, expansionist militants more closely aligned with fascism. The only thing really progressive about the members of the MWBoS is that they saw the value in recruiting outsiders to swell their ranks and dominate the landscape even further. I really hope that Obsidian will develop a game based around Chicago, what they could do with this faction would be incredible.
Fallout 3 was a step forward and backwards in a lot of different ways. It was my introduction to the series, but the more I play and think about it the weaker I feel it is. While I appreciate the gameplay of it a lot more than I do with the old games, the plot-line is about as linear as Tactics while trying to be an RPG. The story had a lot of conveniences, with the Enclave managing to make it all of the way to the Capital Wasteland with more advanced technology than ever before and somehow had a massive swell in numbers (they could have recruited from other bunkers or from a Vault or two but that was never established or explained) and how the Brotherhood of Steel happened to show up there around the same time. I can excuse a fair bit of it, considering how it was Bethesda's first steps into the series but they repeated a few of the same mistakes in Fallout 4.
Fallout: New Vegas is my favorite Fallout game and was a great mixture of the strengths of all of the previous games. They had the NCR, Caesar's Legion, House, and side factions that all represented the struggle between progress, self-destruction, and tribalism really well. The NCR is greedily willing to overextend itself. The Legion is the antithesis to the tribalism that came to dominate the Four Corners after the Great War, destroying tribal identity for the sake of progress but at the same time creating a rigid system and brand new society that will take much longer to destroy or change than it did to create. House represents how an otherwise (relatively) normal person who has seen self-destruction take its course could do whatever it takes, even abandoning a normal human life and lifetime and standing up to the most powerful forces in the post-war world to try to achieve his ends.
This one is put in spoiler tags because it has Fallout 4 spoilers. Be warned.
Spoiler Fallout 4 seems like a very tentative step to try to do a little what was done in Fallout: New Vegas while mainly taking what was successful from Fallout 3. I'm pretty disappointed in how no faction seems to really stand for anything about human nature, and when it does it seems to be too filtered through our modern perspective and in many other ways just repeating from previous games, from Fallout 3 above anything else. The Institute pretty much has the same goals and background as the Enclave, they are just presented as less undeniably evil. They are largely hidden away, ancient, and authoritarian and wish to control the wasteland for the sake of progress or control (I'll have to read up more on them before I can determine one way or the other). The Railroad is an appropriate counter to the Institute's authoritarianism through our perspective. I don't know enough about them to really determine whether or not they actually make sense (I'll have to read up on them to) but from what I can tell, they are just the sort of egalitarian, humanitarian progressives that I do not think make sense in a harsh, wasteland environment, but I'll need more information before I can decide that. From what I can see from human history, people are only so generous when they live comfortably enough that they can spend their time helping others as well and that just doesn't really seem to be the case for places like the Commonwealth. The CWBoS in this game are slightly darker, which I thought was appropriate seeing as how authoritarian systems like the CWBoS was under Lyons usually seems to breed authoritarian types like how Maxson comes across as. Only thing I find weird about them is how their goal seems to be to destroy advanced technologies owned by the Institute instead of hoarding it like the old BoS but that's fine, perspectives and ideas shift a lot. Then there's the Minutemen, who seem to just be fulfilling the purpose of Lyons BoS last time. I'm glad they made them scrappier and more of a pseudo-terrorist freedom fighter group as opposed to the ultra powerful faction that gains a doomsday weapon at the end that Lyons BoS was but they still come across as overly idealistic to me. All of this could have been done just fine if they endings were good... and IMO they were not at all. The endings were vague monologues by the Sole Survivor (the least interesting thing in each of these games is the protagonist so the fact that the ending was him/her talking about how they felt about the events was a massive disappointment). You were told absolutely nothing about what changed about the world or how any of those factions ended up or really much of anything important at all.
And do you know why? So they could allow you to play after the end of the game forever. No major events or changes take place so that you can just keep on playing the game forever. And that's incredibly disappointing to me. /end of post summary rant