A whole bunch of factors.
There really is no one big factor that makes everyone like their favourite generation/game the most/the least. Personal Preference is probably the widest term one can use, but that is way to wide, what influences Personal Preference/your opinion? I will try to look over some of the factors I believe affect this (note, this is highly speculative, I am not saying these are the only factors that influence, or that these apply to everyone, but I think these elements are important when considering anything good or not. Also note, this does not only apply for games, but everything really.) (NOTE, WALL OF TEXT INBOUND)
To begin with, nostalgia can indeed be a very powerful feeling. Almost always we like to reminisce about bygone ages, especially our childhoods when everything was (hopefully) easier, funnier, more lax, etc. If you liked a game before you are likely to remember it fondly. But there are two important factors in terms of nostalgia, when you first played a game and the last time you played since game. I played Super Mario World when I was 5-6 the first time, but was unable to do so as a young teen until I was 13-14. The nostalgia I had was immense. Now I play it every year but I still have strong nostalgic feelings towards it. Nostalgia is the most powerful from ages 5-12, the children years, sure one can find something to be nostalgic towards later, but rarely to the same extent as those early years. And we do not have perfect memory. Chances are if you have not played a game in 20 years, you are only going to remember the most basic aspects of it, and if you liked it or not. So nostalgia is definitely important, but highly subjective.
But there is also the lack of nostalgia when you dislike a game because it is old. If you have played Super Mario Maker with a young person, or any old game for that matter, sometimes they do not like the old games, because they are old. Graphics are important (more on that later) and if you have no nostalgia for a game, people might just opt out of it. This is most likely to happen if there are new versions of the games. I can't bring myself to play Pokemon Red and Blue anymore, despite owning both original, they are just to old and outdated. If there is no nostalgia, you are going to be more likely to be critical of the older, unpolished games, in some cases.
Now I want to talk about the actual quality of the games. This might seem obvious, but a good game is more likely to be beloved than a bad one. A game with the basics down, gameplay, music, immersion, graphics, story, characters, etc is probably going to be beloved. Not by all but having these core things down is what, perhaps not only what makes one good game, but a good series. Quality is really the first step to becoming liked.
Then we have time invested. This might seem strange, but the more time you spend on anything, the more invested in it you will be. Not always, some things keep you in their bubble for far to long, but usually, if you have spent a long time on something, you are going to like it more. This is the most noticeable in long running franchises/games, and why World of Warcraft have devoured every other MMORPG with ease. People are more likely to go back to what they have liked in the past, and spending a lot of time on something will lead you to spend more time on it. Paradox, I know, but it sort of makes sense I hope.
Now I will talk about something that mostly affects new games, expectations. Expectations can make or break anything. If you are superhyped for a game, literally can't wait for it, and it's Ride to Hell: Retribution, you are in for a bad ride. Same goes for good stuff, if you expect garbage and it's really good, you will be pleasantly surprised, and thus like it more. And this of course effects older games as well, but to a lesser extent. It is part of the public conscience that Star Wars 4-7 is good and Star Wars 1-3 isn't, you pretty much know what to expect.
One of the, if perhaps not the biggest reason for choosing any one game over any other, is what you are looking for. Not all games do the same thing, and they do it differently. This is the core of personal preference, simply what you like, and want. If you like FPS games, Pokemon is not a good choice for you, and vice verse (minus Pokemon Snap ofc). There really isn't much more to say here.
Finally, I want to talk about guilty pleasures. This is the most subjective factor, perhaps more so than personal preference. This is the small or big things a game (or anything) do that you like, everything be dammed. Personally, I like Pokemon Black and White the best, despite having played all previous games in the series. Why? Because it was the easiest to train Pokemon. Not EV train or anything, just train, and I loved that. Guilty pleasures can be difficult to pinpoint, but that is really the point, its the games you say "I don't know why I like this, I just do" too.
So there you have it, my take on why people like the games they do. If there is any other reason you have for liking a game, please comment about it, I'd love to hear it!
Sources: My own thick head, the Nostalgia Critic and Zero Punctuation on YouTube, both fantastic series I cannot recommend highly enough.