If I told you two "persons" were in the same body, had the same abilities, are made of the same material, and had the same mind, you'd say I was stupid.
You'd say the "two" persons were actually just one dude.
I would concede, "yes, they're the same nature, and the same being, but they're different persons."
Again, you'd say I was stupid. What's the point of being two persons, if you're obviously only one thing, you may ask.
"See, when they are active, they're one person. When they are simply existing, they're the other. It can be difficult to impossible to tell which person is being displayed at a given time, since there's no evident difference between the two, since they're made of the same stuff, and have the same abilities, as I said earlier."
You probably think this entire argument is stupid, and doesn't make any sense at all. What's the difference between a person and a being? Why should we care which is which, if no one can tell the difference? You would probably argue that they're not two people, it's more like two moods, or attitudes, of one person.
My theoretical dual person is self-evidently absurd, but this is precisely the position of the Divine Trinity celebrated by the majority of Christians. God is supposedly three persons at all time, but the "Holy Spirit" and the "Father" are essentially indistinguishable. They're both omnipresent and omniscient, and have the same intent. It's superfluous to describe such an organization as two persons, especially since the Bible doesn't describe the Divinity as three persons, it's inferred from indirect references to "the Father and of the Son and of the Holy Spirit". There's nothing to say that the Holy Spirit is a person, whatever that means, especially since "Spirit" means "Breath", and has obvious other potential meanings.
It seems to me the "Holy Spirit" is simply the feeling or attitude or ethos that a person has when they are holy. A person infused with the "Holy Spirit" isn't being ridden by a divinity*, they are living the divine life.
This diagram from Wikipedia is self-evidently absurd. Any first-year logic student knows that if A=B=C, then A = C.
The worst part is, none of this effects a single thing. Even the Bible doesn't indicate that belief in a Trinitarian God will affect a part of your life, or determine what happens in the afterlife. All of the endless debates about the nature of God are absurd and semantic, yet people have died for their belief in a non-trinitarian God.
*It's clear the Bible does not think that God is omnipresent, but if God is omnipresent, he can't be more or less with you at any given time.