Q:

What is the noun to describe whether a word is subject or object? 

Just as "number" describes whether a noun is singular or plural, is there a noun that describes whether a pronoun is subject or object?  For example:

The number of the first-person pronoun "I" is singular. 
The __ of the first-person pronoun "I" is subject. 

A:

Thanks to Prof. John Lawler for the very useful comments. 
When I look at it as a question of a hypernym for subject and object in English grammar, the closest that I can find is the concept of case, as currently defined, if not taught. 
ODO (accessed UTC 07:01 today) explains (with English examples):

Nouns and pronouns can be used as the subject or the object of a sentence:
subject --- verb  ---  object
The dog  ---  bit  ---  her. 
…
As can be seen from this example, the pronoun she is used as the subject, but if it is used as the object it becomes her.  These different forms are called cases.  There are three cases in English, subjective, objective, and possessive …

The ODO definition of case, as it relates to grammar in general and not necessarily contemporary English grammar:

4 Grammar any of the forms of a noun, adjective, or pronoun that express the semantic relation of the word to other words in the sentence: the accusative case

I do not find a better taxonomic definition that satisfies the OP's need as of now.