I've always been a big continuity and large/expanded universe guy, so I love it whenever I see that being done in fiction. Not everything has to feel the same way in order to achieve that though. The comics constantly vary in tone and feel yet (mostly) everything is still canon (or intended to be canon in the cases where they contradict each other

). Batman TAS technically had its own feel and identity unique from JLU but it was still very much the same continuity and characters.
Part of that is what carried over to the Nolan films as well. Batman Begins has its own distinct tone and feel as does TDK. However, the story from BB to TDK flows in naturally and despite the tone and feels of the films being different, they don't vastly differ to the point that you question how they could possibly be in the same universe. TDKR also has a distinct feel from TDK but at the same time, I also felt it was a bit too close to TDK in that regard, which I guess makes sense since it tries to tie BB and TDK together. I once saw someone describe TDKR as BB if it was shot like TDK and I think there is some truth to that.
In my opinion, one of Nolan's greatest achievements with his Batman films was finding the perfect balance between continuity and great stand-alone films. I find that other CBM's often struggle with that. On one hand, you have CBM's like the Raimi Spider-Man films that are way too stand-alone and episodic to the point that they age the protagonist too fast and don't leave much room for anything that can flow into the sequel. That was part of the reason why Spider-Man 4 fell through. Then on the other hand, you have CBM's like most MCU solo films that spend way too much time setting up for Avengers and future sequels that they just scream of a sequel and cannot be judged alone. For the latter scenario, it usually leads to opinions on the first film being altered down the line due to opinions formed from future films. I hate to say this since I loved The Amazing Spider-Man and Man of Steel but they are guilty of this as well in the same way the MCU films are. They intentionally leave too much for the sequel that my overall opinion of them will be affected by what the sequels do. Nolan's films didn't really have that problem.
Since TDKR is a very controversial film in this thread, I am going to stick only to analyzing Batman Begins and The Dark Knight in the following paragraph in order to keep as many people as possible on the same page.
Both BB and TDK had endings that didn't need sequels. Batman Begins isn't just an origin story; it stands on its own. Had TDK & TDKR not have existed, it still would've been a great conclusion to that Batman. The same thing can be said about TDK's ending. In fact, TDK's ending was so good that one of my biggest fears was that they wouldn't have found any good way to top that ending in future films. BatLobsterRises brought up a great point when he said that regardless of who you are, whether you thought TDKR honored or went against TDK's ending, TDK's ending was an absolutely fantastic ending for all of us. And yet, there was still room for more stories after both BB's ending and TDK's ending. He focused on the movie at hand - "one film at a time" is his motto according to Goyer

- and didn't let the future bug him too much to the point where he made movies that would still work well even if they never had sequels. At the same time, he also left little things here and there that have set up things for the future of the franchise (Joker's card at the end of BB, Scarecrow on the loose, the rise of "freaks", Batman on the run, etc.) and never let the time gap in between the movies fast forward too much (excluding the 8 year gap in TDKR). This established a cohesive ongoing continuity out of films that could be looked at almost completely individually. This is how CBM's should be done IMO. Maybe this balance was found intentionally or maybe they just got lucky due to the Nolan franchise being made at the exact time in which CBM's where transitioning from the way-too-stand-alone-with-no-setups style to the more one-episode-of-the-bigger-picture type of style. Either way, they managed to pull it off.
It is interesting how so many fans that claim to have been dissapointed with TDKR also claim to still like BB and TDK just as much as before. Some even say that TDKR was so bad that it made them like the first two a lot more (which I personally think is a bit of a ridiculous stretch). How many people would have felt that way had these films followed the MoS/TASM/MCU route when it came to film continuity which, as I said before, planned too much for the future that it took away from the movie at hand? Imagine if BB and TDK did this and people walked out of TDKR with the same opinions they have now. Of all the people dissapointed with it, how many would have had that dissapointment affect their opinion on the first two films? Would the same people still say that they still like BB/TDK just as much? Would the same people say that they now like BB/TDK more than they did before? That is something to think about. Then again, the same thing can be argued with those that were dissapointed with BB and/or TDK who loved TDKR. Hence average movies like Thor and Iron Man 2 being put on a pedestal due to the quality of The Avengers.