Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
Honestly, I think you nailed it. I was just reading some Saint Severus last night and then some Heinrich Zimmer on Brahmanism earlier today, and also yesterday my priest and I were discussing the Holy Spirit in the context of the thread I posed about the "Life-Giver"
Since we attribute differing personalities unique to each of the Persons of the Trinity, I'd say its safe to at some level suggest that The Father, the Son, and the Holy Spirit are completely different individuals just as we human beings are all essentially human (thus one) and yet clearly individual souls/conscious entities. However, here is where I would subtly avoid heresy, our own human nature, though unifying as one essence of humanity, is also by nature splintering and chaotic, rather than creative and unifying. It seems quite natural then in time that human beings would splinter off from one into two and four and ten and a billion etc etc..
However the nature of Divine is unifying, life-giving, creative, and so while each Person, due to their own hypostatic manifestation in the Trinity has their own respective personality, still because their own inherent Nature is Divine (unifying) they remain Eternally (that means before, during, and after) One. Think of Nature and Hypostasis as synomous with the concepts of function and form, the function is the nature/essence and the form is the hypostatic manifestation the way the function of a chair is "sitting down" and its form takes on a manifestation able to sit on (in actuality anything can become a chair if its form allows it, and with God His function is Divine and so any form that can be Divine is Divine, that is, any of the Three Persons of the Trinity is God). Our nature/function is to be human beings, and our form is our bodies/souls. The function/nature of the the Divine is to Self-Exist, Create, and Sustain and we know through the Mystery of the Holy Trinity that the Form/Hypostatic Manifestation of God is in Three Persons.
While the "race" of men is inherently divisive, the "race" of the Divine Nature is essentially unifying and thus eternally unified. The fundamental flaw of Hindu theology is that is mistakes human nature as somehow being in the same category of oneness as Divine. Yes, fundamentally we can say were are all "human" and this is a unifying reality, but this is more so an ontological rather then theological analysis, an almost "duh" kind of thing. Of course because we are all human that makes us all the same, but being the same hardly makes us One in the sense of pan-unity. When Hindus attribute the Oneness of Humanity and actually all-matter and things to the Self and assume this Self is the One God (Source, Creator) they make the mistake of not separating the Divine from the created. The Divine is indeed One, and it is indeed the source of all things, but that only makes all things one philosophically, not ontologically, clearly in our tangible world there is a separation, a distinction, an individuality. The Hindu metaphor that all pots made of clay are unified as being "clay" by essence even if they break and are no longer pots, fail to see that humanity, while one in essence, does not automatically make us ontologically one, that is to say, actually
one and not individualized. Only God is One because only the Divine Nature is One, the natures of created things do not necessarily have this same Monad existence.
So again, with the Trinity, I would say that each Person does have a respective individual Personality because of their Own Hypostases, and yet because They share the same Divine Essence, this Divinity is unifying and makes the Individuals truly One and not just essentially like the rest of our own understanding of ontological oneness in creation. Human beings are one in the sense of being "human" but remain individuals in reality be it spiritual or physical, where as the Divine Persons are individuals in hypostasis but their Divine Nature has the function and essence of Unifying Oneness which brings them Eternally Together. If the Trinity were not Divine, then They would be fully individuals like human nature has ourselves, but the Trinity are Divine, which is really altogether Inconceivable, Immaterial, Unexplainable and so it can easily be both individualized and yet harmoniously One, its a Mystery