Tigers (Osaka Tigers: 1935-1939, 1946-1960, Hanshin Tigers: 1961-present)

The Hanshin Railway Company, which owns Koshien Stadium, established the Osaka Baseball club on December 10, 1935. They became one of the original teams of the Japan Professional Baseball league in 1936. The nickname "Tigers" was modeled on the Detroit Tigers, since the Hanshin industrial area around Koshien resembles one of the America's biggest industrial zones, the Detroit industrial area. The Tokyo Giants took their name from the New York Giants so Hanshin took their name from the Detroit Tigers.
The nickname, Tigers, was used independently on tickets and posters --- "Tigers vs. Kyojin" and "Tigers vs. Hankyu" --- until it was abolished in fall 1940, due to World War II.
In 1946, the team became well known as "Moko" (a fierce tiger written in Chinese characters). That was during the period when sportswriters used Japanese versions of nicknames like "Kyojin" (meaning "Giants" written in Chinese characters). You can see those nicknames on most magazines during that period.
However, the parent company, the Hanshin Railway Company, requested them to use their name "Hanshin". Thus, the nickname was gone within a year. The parent company might do that in rivalry with other teams but if they kept the nickname, they would have been given special treatment like the Yomiuri Giants today.