Here’s what I mean. The Bible clearly teaches that the Father is God, that Jesus is God and that the Holy Spirit is God ... the “Trinity.” Yet it also teaches that there is only one God. Explain that if you can.

I will not pretend to offer a comprehensive account of this doctrine.

And I agree that it is "mysterious." I also believe that it may be perfectly rational to believe something that remains mysterious.

But I would emphatically insist that it is never rational or virtuous to believe something that one takes to be or entail a contradiction.

If the Christian doctrine of the trinity entails a contradiction, then it is necessarily false and worthy of the scrap pile--or else display in the museum of crazy ideas.

You are correct that the popular analogies fail. The "water" analogy would imply modalism--the idea that each of the "persons" of the trinity amounts to a different form that the one person who is God might take: Father, Creator, Redeemer, Comforter, etc. The "egg" analogy also fails. None of the parts is fully an egg.

The closest I've come by way of analogy is marriage--my marriage in particular. I am conscious of a kind of identity between myself and my wife so that my own self concept somehow includes hers, and vice versa. Yet we are also two separate persons. But this, too, fails, if for no other reason than the sheer contingency of the identity that I think I discern.

But I think a bit may be said in reply to your question quoted above.

Bill Clinton once said "It depends upon what your definition of "is" is." The same is true here. There are at least two uses of "is" in our language:

The "is" of identity: Mark Twain is Samuel Clemens. Here, we are asserting that the thing referred to on the left side of the "is" is one and the same thing as what is referred to on the right.

The "is" of predication: Mark Twain is an American author. Here, no identity is asserted. Rather, to the left is the subject and to the right is the predicate.

Confusing these two senses leads to logical chaos. Consider this argument:

(1) God is love.
(2) Love is blind.
(3) Ray Charles is blind.
(4) Therefore, Ray Charles is God.

Similar problems arise in thinking about the Christian doctrines of trinity and incarnation. So consider:

"Jesus is God"

Is this an identity statement or a predication? If it is an identity statement, then we have your problem. Because, given, "The Father is God" and "The Holy Spirit is God" we can neatly derive:

"Jesus is the Father"
"The Father is the HS"
"The HS is Jesus"

And this, of course, would be to deny that difficult doctrine of the trinity (and lead to modalism).

Perhaps the assertions are best understood as predications. No one would think that "Mark Twain is an American author," together with
"Garrison Keillor is an American author" entails "Mark Twain is Garrison Keillor."

So perhaps we should take your statements as predications whose import is something like:

"Jesus is divine"
"The Father is divine"
"The HS is divine"

The trouble, though, is that this can begin to look like tri-theism. How is this different from the following?

"Zeus is divine"
"Athena is divine"
"Poseidon is divine"

AT this point, I'll simply note two things. First, the classical doctrine of the trinity maintains that, in God, we have three persons (hupostases) but only one substance (ousia) or "unit of being." It would be a formal contradiction if it affirmed that God is both one and three persons or both one and three substances (at the same time and in the same respect). But it does not affirm this. Rather, it is the assertion that, unlike you and me, where we find a sort of one-one ratio between persons and substances, in God things go differently. There, the person-substance ratio is three-one. This is mysterious. Indeed, I think it is weirdly mysterious. But there is nothing formally contradictory about it.

Second, not a little attention has been given to this topic by philosophical theologians. The project is to offer a model on which the doctrine is coherent. A rather large contingent of thinkers opt for what is sometimes called Social Trinitarianism, and their task is to offer a model that does two things. First, it preserves the ontological distinctness of the persons of the tritnity, just as you and I are ontologically distinct. Second, it "tightens up" the relationship among those persons in a way that avoids mere tri-theism or polytheism. And so there is talk of "necessarily harmonious wills" or the ways in which the individual essences of the persons mutually include or entail one another. No one I know who engages in such work thinks that he or she has nailed the doctrine down. Rather, the challenge is simply to ask, and answer, "Is there any possible way of understanding this doctrine that is both consistent with the doctrine itself and logically coherent?" There are several promising models.

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"Puddleglum" by Weatherwax (one of the Muddlings).

Jeeves to the Rescue