The grammar I was taught in 4th grade says that "is" does not take an object, but rather a predicate nominative (PN, as in "she is a doctor") or predicate adjective (PA, as in "she is friendly").
Predicate nominatives take the subject case, not the object case. This rule, of course, results in the question that is more common than "why don't we say, 'who is her'?", which is whether we should say "it is I" or "it is me"? The grammatical pedant will typically advocate "it is I", though I don't think many would say with a straight face, being asked whether a group of people is the same group they saw the night before, "yes, that's they."
In light of the comments and other answers, it seems a good idea to refer to some sources. Wikipedia(https://en.wikipedia.org/wiki/Predicate_(grammar)#Predicates_in_traditional_grammar):
A predicative nominal is a noun phrase that functions as the main predicate of a sentence, such as George III is the king of England, the king of England being the predicative nominal. The subject and predicative nominal must be connected by a linking verb, also called a copula. A predicative adjective is an adjective that functions as a predicate, such as Ivano is attractive, attractive being the predicative adjective. The subject and predicative adjective must also be connected by a copula.
This leads one to https://en.wikipedia.org/wiki/Linking_verb:
In traditional grammars and guide books, the term linking verb is used to refer to verbs that describe the subject or link the subject to some complement such as a predicate adjective or predicate noun. This includes copulas such as the English verb be and its various forms, as well as verbs of perception such as look, sound, or taste and some other verbs that describe the subject, such as seem, become, or remain. In addition to predicate adjectives and predicate nouns, English allows for predicate prepositional phrases as well: John is behind the cocktail cabinet.
The following sentences include linking verbs.
- Roses are red.
- The detective felt sick.
- The soup tasted weird.
- Frankenstein resembles a zombie.
This explains why one should say "that music sounded awful" rather than "that music sounded awfully." Or, as I once heard someone say, or words to this effect (I sadly cannot find it online):
A wet dog smells well, but a wet dog smells bad.
Contrast the definition of "object" (from https://en.wikipedia.org/wiki/Object_(grammar)):
Traditional grammar defines the object in a sentence as the entity that is acted upon by the subject. There is thus a primary distinction between subjects and objects that is understood in terms of the action expressed by the verb, e.g. Tom studies grammar - Tom is the subject and grammar is the object.
Linking verbs such as "to be" do not act on one of the two nouns in a simple sentence. Rather, they describe an existential relationship between them.