When and how the study of naming objects began in philosophy? I mean how objects are named, how a same object can have different names refering to different aspects or features of the object.
The closest thing to the "study of naming objects" today, it seems, would be semiotics or semiology. As a sustained field of interest it has begun with Ferdinand de Saussure at the beginning of the 20th century. He convincingly and very influentially argued that language is a system of signs and that it does not pertain to "the world in itself" but is rather arbitrary.
Of course, Saussure had many predecessors interested in the very same question, but his posthumously published lectures were the main impetus behind turning this interest into a full blown discipline (which today one will find very far away from the original Saussurean conjectures).
In Cratylus, for example, Plato is already pondering whether the names of objects pertain to the "essence" of these objects or if they are simply arbitrary. Plato in this particular case is either on the essentialist side or takes no position. Aristotle, on the other hand, argued for conventionality of language: if a word had some sort of intrinsic connection to what it signifies why, then, even such basic concepts as bread and wine are called differently in different languages?
"How objects are named" is something of a mystery to me. I imagine it's been a puzzle since Adam and his garden full of animals for naming. So to speak.
For "how a same object can have different names referring to different aspects or features of an object", the most thorough treatment you'll find first is in Frege. If there are earlier mentions of this hydra, you'll find them there.