Do Tomb Raider: Legend, and Tomb Raider: Anniversary use the same game engine? The character models definitely look the same.
Legend's has a feature called "Next-Generation graphics." Is there a way to turn it on in Anniversary?
More or less same graphics engine. More or less the same player models. But all the levels are different (more or less the levels from the original game, massively updated with new technology and tricks they just couldn't pull off in the old engine) and Lara has a slightly different bag of tricks (pole balancing, assisted wall running, etc).
Calling it "the same game" is doing both a disservice.
As said, both uses the same engine but Legend came first than Anniversary so you can expect a few aces not available on Anniversary that are present on Legend (i.e. next generation content).
Maybe you don't know this but Anniversary it's a remake of the first Tomb Raider, so yes, it use the same Legend model for Lara, but not so much for enemies.
So far, the only main differences I have noted is that Lara doesn't have a flashlight in Anniversary, nor binoculars, she can't stand top of swing poles and jump from them, no PDA (you have a journal instead, but it's far less useful), no bike sequences, magnetic hook in mansion looks more like a grappling hook even when it do the same, next generation content can't be enabled, PC port is badly optimised, as consequence the game will run worst at same settings used in legend. Besides that, it's pretty much the same game mechanics...