A few years ago, one of the rabbis in my yeshiva spoke about this and the explanation he gave was more philosophical than the options you presented. According to him, the statement is more about your inner identity, what makes you you.
He brought by way of explaining one of Rabbi Nachman's stories:
Ivan the Coachman:
A wealthy man went once to the city with his coachman, Ivan (Ivan, in Yiddish, has the connotation of being a coarse, mundane sort of person). Upon arriving, the man left his coachman on the side of the road and went to do his business. Along came a policeman and confronted Ivan by saying: "Ver bistu!" (meaning, "Who are you? You can't be here!)
Ivan, however, thought the officer was asking for him to identify himself, so he replied: "Ivan!"
Incensed, the officer shouted at him: "Ver bistu!"
And again Ivan mistook his meaning and answered: "Ivan!"
The officer blew up and began to beat up Ivan. Ivan escaped into a nearby alley until his boss returned. When the wealthy man came back, he called to Ivan: "Ivan!"
Ivan, cowering in the alley, whispered to his boss: "Here in the city, you can't call me Ivan. You need to call me Ver bistu."
When they left the city, Ivan turned to his boss and said: "Now you no longer need to call me Ver bistu; you may call me Ivan." (Hebrew version, translation based on this video)
The story is a parable: each and everyone of us is Ivan. The policeman is the world. The world is constantly demanding of us to have an answer to the question: "Who are you?" Whoever doesn't know who he is, won't know what's the right place for him in the world. And if you're in a place that's not right for you, you'll suffer, like Ivan was beaten up. If you do not know who you are and what you are doing in this world, you cannot live a meaningful life and be an integral part of the world. In the site that brings the Hebrew version, the rest of the parable is also explained, but my rabbi drew focus on to this part.
The Kotzker is relating a similar idea: If you are you simply because you are entity A and the other guy is entity B, then you have no deep, personal identity. You have no meaning to your life, no particular reason for existing. But if you are you because you have a special, personal identity, and he's himself because he has his own special, personal identity, then you are you, you are a special person with meaning to your life and a reason to exist and he is himself with his own specialness and reason for existing.
I will add that another related idea is what the mishnah in Sanhedrin 4:5 says:
"וּמֶלֶךְ מַלְכֵי הַמְּלָכִים הַקָּדוֹשׁ בָּרוּךְ הוּא טָבַע כָּל אָדָם בְּחוֹתָמוֹ שֶׁל אָדָם הָרִאשׁוֹן וְאֵין אֶחָד מֵהֶן דּוֹמֶה לַחֲבֵרוֹ. לְפִיכָךְ כָּל אֶחָד וְאֶחָד חַיָּב לוֹמַר, בִּשְׁבִילִי נִבְרָא הָעוֹלָם." - "but the King of kings, the Holy Blessed One, has stamped every human with the seal of the first man, yet not one of them are like another. Therefore everyone must say, “For my sake was the world created.”"
Each and every one of us is in this world for his own special purpose. But the first step towards finding and realizing that purpose is understanding who you truly are.