This article is devoted to analysis of current concepts in modern times "Language Personality" and "Personality". The text of the article provides information on how to treat these notions in terms of the sections of linguistics as sociolinguistics, psycholinguistics, lingvosocionics, cultural linguistics, etc., As well as from the point of view of psychology. In addition, the article focuses on the study and modeling capabilities and linguistic identity of the person on a material of a literary text. The basis for the proposed variant modeling personality served as a theoretical model of personality structure proposed by Sigmund Freud, which consists of three elements: the Id (the unconscious beginning), Ego (the conscious part) and Superego (level representing the social norms and rules of behavior in society). Examples given in the article of the literary text by D. Granin "Zubr" represented the possibility of a literary text`s using as a material for the language personality`s modeling.