What the writer means with "identity" is that sentences of the type
The thing A is a thing B
(eg: An apple is a fruit. A dog is an animal.)
contain the same idea as in mathematics 2 + 3 = 5.
Sentences with a form of 'to be + subject complement'
totally differ from sentences with transitive verbs (verb + direct object).
I don't think that "identity" is optimal to describe the relation between subject and subject complement. Actually I think a good term is lacking. Even "equality" is not appropriate in all cases. If I say "The child is sick" then I would not say there is an equality between 'child' and 'sick'.
The formulation "to be is an identity" is totally confusing and a "non-information". The traditional term is 'linking verb'. The Latin term is copula meaning ribbon. The verb 'to be' is a mere abstract link between subject and subject complement (a noun, an adjective, an adverb) and has the same function as the sign = in mathematics. And with subject complements 'to be 'is a full verb, and not an auxiliary verb. Auxiliary verbs help to form tenses and similar things.
You can say that the complement after 'to be' gives some information about the subject. We could also say the linking verb 'to be' describes a relationship between subject and subject complement (also called predicative complement).
In a sentence with a transitive verb we have a different situation. A verbal activity goes from the subject over to the direct object. Latin trans-ire is 'over' + 'to go', and 'transitive' means "going over".
Actually a good and easy term for sentences of the type
- A dog is an animal - The child is sick - He is here -The book is on the table
is lacking. My personal term is simply "is-sentences", defined as a form of 'to be + subject complement'. It is astonishing how many learners wrestle with is-sentences and "transitive" sentences, obviously because grammars neglect this side of language and don't offer appropriate terms for describing language.