The first thing that comes to mind is
Earth Defense Force, it encompasses the entire planet but most of the series
takes place in urban areas of Japan which are entirely destructible and often under attack bygiant bugs and monsters.
Another title that comes to mind is
Robot Alchemy Drive, by the
same developer as EDF (Sandlot), it also happens to take place in Japanese cities and has a far more anime approach to its story and characters, you control giant robots in this game using every button in your PS2 controller, it takes a bit to get used to but its extremely satisfying to play.
Natsuiro High School for the PS3/PS4 is a game with an approach similar to that of Shenmue, focusing on a day to day's student life with a very unique mechanic that involves underwear and photo cameras
This game is sadly unavailable outside of Japan but getting a hold of it these days via other means should be relatively effortless, it's quite reminiscent of Akiba's Trip except it's a laid back experience.
Tetsujin 28-Go is based on the eponymous manga series, set during post-war Japan it offers a slightly different approach to Japan as a setting than the one you would
Gokinjo Boukentai is a cute little JRPG set in urban Japan reminiscent of MOTHER, if I am not mistaken it recently got a fan translation. I haven't played much of it nor have I touched this in several years but it definitely "felt" like Japan so to speak and it had really cute visuals.