Demonstrative pronouns in Italian are "questo," corresponding to "this" and "quello," corresponding to "that" in English, which also have their plural counterparts, "these" and "those." In my language, which is Georgian, there are 3 types of demonstrative pronouns according to the number: 1st person: რა არის ეს? What is this (that I am holding)? 2nd person: რა არის ეგ? What is that (that you are holding)? 3rd person: რა არის ის? What is that (that he/she is holding)? So English uses "that" for 2nd and 3rd person while the Georgian language has different demonstratives for 2nd and 3rd persons. My question is this: Does the Italian language have a specific demonstrative pronoun apart from "quello" to indicate the thing the person being spoken to is holding, or does it use "quello" for both 2nd and third person pronoun like English does?
Sounds like you are looking for codesto, nowadays rarely used outside of Tuscany:
codésto (o cotésto) agg. e pron. dimostr. [lat. eccu(m) tibi iste]. – Indica persona o cosa vicina a chi ascolta, o a lui relativa, o nominata subito prima.
That is, codesto is an adjective or a pronoun that indicates a person or an object close to the listener, or related to him, or something that has been mentioned (by the listener) immediately before.
Outside of Tuscany, usually the role of codesto is played by questo or quello, depending on whether the object referred is close to or distant from the speaker (so that the position relative to the listener is irrelevant).
For instance (these examples are taken from the book Italiano by Luca Serianni), in Tuscany one can say
Sono nuovi codesti guanti?
referring to some gloves worn by the listener. But in the rest of Italy one would say questi guanti (assuming that the interlocutor is close to the speaker in a real or ideal way) or, if there is the possibility of some misunderstanding, questi tuoi guanti or simply i tuoi guanti.
No, Italian does not have different demonstrative pronouns according to the grammatical category of person, but only along the parameter of proximity of the referent to the utterer and the addressee ("questo": close to the utterer, "quello": far from the utterer (and also in some cases from the addressee").