Samanya adhikaranam
Are all the words in the sentence referring to the same thing or do they refer to different things. Since we are discussing non-dual reality and that You are Brahman, then ultimately we are talking about one reality. The words all point to the same reality, same locus (adhikaranam). E.g. blue lotus, the word lotus and the word blue both refer to that same flower.
Visheshana Vishesha Bhava
Is the meaning of the words a noun- adjective relationship, substance (Lotus) and quality (blue). Or is it noun to noun? Ishvara (God) and Jiva (You) are one (connection to infinity). This noun (infinite Ishvara) to noun (limited Jiva) as a direct meaning excludes other names and forms, that Ishvara (not Vishnu, shiva etc) are Jiva (you or me). However, such direct meaning also has glaring contradictory qualities and are equitable (Ishvara are omniscient etc and Jiva as limited). Therefore, we need a 3rd step of investigating the implied meanings.
Lakshya (pointed/implied) lakshana (pointer/defn) bhava:
What is the Tatpariya/intended and Implied meaning of That and This after discarding any “qualities” superimposed on them. Eg. that devadatta (young one from different city) is this (elderly, coming towards me). Any contradictory qualities (include space-time effected) are discarded and the intended meaning “person” is understood. Eg. That blue lotus (month ago in the pond) is this blue lotus (aged and in a pot). or Ganga river implies the same river whether it was a trickle in the Himalayas or flooding at the mouth in bangladesh.
So now analyzing the sentence Tat Tvam Asi. Tat is implying to Nirguna Brahman, pure consciousness without the upadhi of universe created by maya. Tvam – Nirguna Atma – Self Consciousness without the upadhi of body-mind-senses. Asi – The Consciousness substram is non-dual, one without a second, the same. The sentence is before the universe was manifested it was one existence without a second.
However, Nirguna brahman is beyond language of words. So a word (written on spoken) has a direct meaning (through pratyaksha pramana through vachya artha), it must refer to something within this manifested union with this universe ( created by maya), it could be the cause of the universe Ishvara, because cause and effect are manifestations in the same order of reality. The range of a language can only indicate manifested cause and effects and therefore it can only directly denote Ishvara, as saguna Brahman. However Ishvara (all pervading, etc) and Jiva (limited, insecure) can never be equated. So there has to be deeper meaning to the words.