Biological key file format
I searched a bit and could not find any such format, but I am not a specialist in taxonomy.

From what I understand, the structure you want is that of either a tree or a DAG (Directed Acyclic Graph). Both are straightforward to store in a variety of formats.

One possibility is to use a format similar to other biological data that has a similar data structure. For example, Gene Ontology is a widely used dataset that has the structure of a DAG and has a number of formats including XML listing the parents and additional data of each node in the graph. See here for their formats.