When Cate awakens Parker, who has fallen asleep watching Game 6 of the 1996 World Series with Hachi, the footage shown on the television is of then-Yankee catcher Joe Girardi running out his run-scoring triple in that game. Presumably it's late, and Parker and Hachi had been watching the game quite a while for Parker to have fallen asleep, but Girardi's triple came in the 3rd inning. It's also questionable whether Parker, who's depicted as an avid Yankee fan, would fall asleep during a World Series-clinching game.
(at around 32 mins) A girl dressed in light color coat and jeans with a black handbag walks toward Mary's bookstore. In the next shot the next she has just opened the gate of the station and walked out, then she walks down the road again before Hachi welcomes Parker.
The commuter train that Parker rides is equipment (locomotive and cars) of the Providence and Worcester Railroad. However, the Providence and Worcester does not operate regularly scheduled passenger trains and did not at the time the movie was being made.
When they confront the skunk in the backyard the scene ends with the wrong assumption that skunks spray backwards which is untrue. Skunks will stand on their front paws, arch their backs and spray forwards over their heads.
When Hachiko has moved in with Andy, Michael and Ronnie (still a baby), he is shown laying on the floor, while Ronnie collides with the dog repeatedly in his baby walker. In the first shot showing the walker from the side, the baby is absent from his walker. Cut to a master shot showing baby Ronnie in the walker, followed by another shot of the walker colliding with Hachi, similar to the first which shows now the baby's legs.