What is the study of names?

Answer 1

The study of both plants and animals is referred to as Biology.

that can be also narrowed down to botany for plants and zoology for animals.

Answer 2

Onomastics or onomatology is the study of proper names of all kinds and the origins of names. The word is Greek: ὀνοματολογία (from ὄνομα (ónoma) "name")). Toponymy or toponomastics, the study of place names, is one of the principal branches of onomastics. Anthroponomastics is the study of personal names. For some cultures (and celebrities), only one name is needed to indicate a certain person without ambiguity. In others, a single personal name may be insufficient, requiring alteration to a hypocoristic/diminutive nickname or addition of a byname based on a specific individual's traits, family, home, occupation, or other. In most of the world, individually-based bynames have become hereditary family names, perhaps retaining little descriptive resemblance to the ancestral namesake's original byname. Most Western European cultures use the name order indicated by the common synonymous phrases "first name" for personal name and "last name" or "surname" for family name. However, this differs from traditional East Asian and Hungarian usages, which place the family name before the personal name.