Analogies certainly do have their uses.
I wanted to further respond to part of the original post.
Bill Clinton wasn’t being entirely non-sensical when he said, “It depends upon what the meaning of the word ‘is’ is.” When speaking of the Trinity, the word “is” should not be construed as mathematical equivalency or mathematical identity. When speaking of the Trinity, stating that “The Father is God, The Son is God, and the Holy Spirit is God” should be understood as meaning “The Father entirely possesses the one nature of God, The Son entirely possesses the one nature of God, the Holy Spirit entirely possesses the one nature of God.” But the Father is not the Son or the Holy Spirit, and the Son is not the Holy Spirit, and there is only one nature of God.
Admittedly, this is still not something that the human mind easily wraps itself around. But it’s not mathematical nonsense we are speaking here. The hurdle is in understanding that there is a distinction between nature and person even amongst the things we are familiar with such as you and I. It is then understanding that our common sense notion that there is a one-to-one correspondence between nature and person is not a law of logical necessity. Since there is no logical necessity that there must be a one-to-one correspondence between nature and person in all things (and that ratio is only a matter of common experience of human persons), there is no logical contradiction inherent in stating that there is one nature who is three persons or three persons who possess one nature. Mind-boggling? Yes. Logically contradictory? No.
The Trinity is not something we would be able to deduce only from our common experience precisely because there is nothing else like it that would lead us to posit it. We accept it because it is a direct revelation of God into who He is personally.