Taro appears 66 times at rank 1582 on neutral-gender pets. It's a Japanese given name with deep cultural roots: Taro is traditionally the name for a firstborn son in Japan. American pet owners use it primarily for its sound and its association with the purple-grey taro root, now a food-culture aesthetic fixture.
Two Taroes at Once
The traditional Japanese given name Taro (太郎) means "great son" or "firstborn" and has been a common masculine name in Japan for centuries. The taro root is a starchy Asian vegetable whose purple-grey color has become a social media food aesthetic, with taro milk tea and taro ice cream driving an Instagram-fueled fan base. Pet owners are usually thinking about the food trend, though the Japanese cultural layer arrives regardless of intent.
Sound and Breed Fit
TAH-ro is two syllables with a clean, open sound that works equally on dogs and cats. Shiba Inus and Akitas fit the Japanese origin with obvious cultural coherence. Blue British Shorthair cats fit the food association. The human name connection is at /names/taro.
The Counter-Reading
Taro sits at a crossroads of cultural sources, and owners usually aren't fully aware of all of them. That's fine. The name works on sound alone, and the layers of meaning are a bonus rather than a requirement.