It seems to be more common to "reboot" a franchise or series than it does to tell a serialized story. Even when a story directly continues in a sequel, it often stands largely on its own and does not call very much attention to the games that came before it outside of recurring characters or off-hand references. Uncharted comes to mind - you're always Nathan Drake and his merry band of thieves and adventurers, but each story is mostly self contained.
There are, however, the occasional standout franchises or long running series. Among all games that treat their canon with respect and expand upon it well, Metal Gear Solid as a franchise, but especially the second game, is the king for me because of its added meta-narrative. What follows is a spoiler-free synopsis of why I think this is the case.
MGS2 goes about deconstructing the idea of a perfect sequel. You begin as Snake in the Tanker, and it has all the makings of a by-the-books sequel to the previous game - an older, more grizzled Snake, a bigger and badder Metal Gear, some returning names and villains from the first game... until it drops you off a couple of hours later as a new character. He is Snake in every way, but he also isn't. His code name is Snake, but it's quickly revoked. He's an enhanced supersoldier like the Snake we know, but unlike Solid, he's a newbie. He's obedient and capable, but he doesn't have the cool and composed air that Solid Snake did, instead frequently revealing his nature as a rookie on the field and losing his cool or making outbursts towards others. He's fighting an organization similar to FOXHOUND, but unlike FOXHOUND's clear-cut demands of a large sum of money and the corpse of a genetic supersoldier, their true motives and capabilities seem cloudy by comparison.
The villains and the skeletal structure of the narrative are deliberate retreads of the first game, in a much more hamfisted way than the tanker portion of the game. This framework all comes to a head during the endgame, when the player and their own expectations are used to shatter their perception of what the game has apparently been trying to do until that point. Not only that, it then shatters those expectations with twist upon twist upon twist. What is the S3 plan? Even after finishing the game, you might still doubt the validity of what you've been told.
Even beyond the meta-narrative, and the fact that it so brilliantly uses its own medium to drive home a point, there is a legitimate extension of the world introduced to us so far. We learn of other organizations similar to FOXHOUND, the impact of Metal Gear on the world at large is addressed, Solid Snake gets a proper continuation of his story and grows quite a bit as a character, and other characters, such as Otacon or the colonel, return or get properly alluded to. This all happens while introducing us to new characters and new stories, even when those characters may initially feel like retreads of the previous game. It really is - despite its deconstruction of the concept - the perfect sequel in my eyes.
Metal Gear Solid 3 carries on the notion of building a cohesive world even more by going back in time to reconcile the narratives of the Solid games with those of the original two Metal Gears by introducing us to Solid Snake's "father," Big Boss. Up until this point, it was a bit of a hard sell that the noble Snake (and even the not so noble Snakes) were clones of this man, but by going back in time and seeing the physical and ideological resemblance, it helps the player accept Big Boss's relationship to his sons, and his fall from grace, in a very organic way.
You can see which pieces of the father were inherited by which son, and Big Boss himself, known during this time as Naked Snake, feels like a fresh new character despite being a literal and figurative clone of those who came before. This Snake grew up in a more natural environment, and, at the time the game takes place, is not a genetically enhanced or otherwise special soldier. He is simply a man on a mission, and his character reflects that. He jokes around with his support team, he smiles, and he expresses self-doubt, all while remaining a rock solid agent that's willing to get the job done. Metal Gear Solid 3 uses its protagonist to create a perfect fusion of the old and the new.
I think the entire Solid series, outside of the subsequent Big Boss games (Peace Walker and The Phantom Pain), is one of the best examples of building a strong continuity between games that there is. Even from a gameplay standpoint, you watch the games slowly evolve; everything is familiar enough to be the Metal Gear you love, but different enough to be a fresh experience. MGS1 has the stealth based gameplay and the soliton radar. MGS2 keeps these, but adds dynamism to its encounters by having bodies stay on the ground and soldiers call for backup. MGS3 take the idea of focused infiltration and sticks it in a wide open jungle setting, while also removing "future tech" such as the soliton radar and replacing it with 1960's technology. MGS4 has quality of life improvements such as crouch walk and a new camera. Everything about these games feels distinctly "Metal Gear," even when it's a new idea, and that is why I think this series truly serves a shining example of maintaining an excellent continuity between entries.