NO, it is not finished.
The anime finished after the manga:
Anime finished: April 01, 2004
Manga finished: November 01, 1991
... But MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Anime - Base
Release date: August 18, 2003
Anime - Sequel
Release date: March 27, 2004
Massugu ni Ikou. tells the heartwarming story of Mametaro, a talking dog who experiences the world in a unique way.
Though only he and his fellow animal friends can understand him, Mametaro navigates the everyday life of his owner, Iku-chan, and Iku-chan's girlfriend, Hanako-chan.
He's a loyal and loving companion, but isn't so fond of Iku-chan's friend, Akiyoshi.
The anime explores the ordinary neighborhood through Mametaro's eyes, transforming it into a vast and exciting landscape filled with adventure and the simple joys of spring. It's a story about friendship, loyalty, and seeing the magic in the mundane from a dog's perspective.