China as a pet name operates on multiple levels simultaneously: it's a country name, a word for delicate porcelain dishware, and a color reference (blue-white, porcelain-pale). Which reading the owner intended depends entirely on the context, but all three produce interesting results on the right animal.
Porcelain Logic
The porcelain reading is probably most common for pet owners — China as the delicate white dishware that gets put away for special occasions. Applied to a pale, refined animal, it carries a sense of something precious and slightly fragile. That irony works differently on a 60-pound dog than on a cat, which is why China skews toward female cats and small female dogs in registry data. The contrast between "delicate china" and a real animal's actual behavior is part of the warmth.
Geographic and Cultural Dimension
China as a country name gives the name a geographic-naming option for owners who choose city and country names for their pets. It sits in the same category as Cairo, Rio, and India as a place name with character beyond its geography. For animals with connections to East Asia, there's a more personal resonance possible.
Sound Profile
CHY-na: two syllables, the CH opener is soft, the long I vowel carries, -na ends warmly. The name is easy to call and easy to recognize. Chinese Crested dogs named China are an obvious match; so are pale-coated cats. Compare with Jade for a different East Asian reference point with a more modern feel.