So a few notes on your terminilogy in your question from the perspective of the orthodox, canonical viewpoint. As otherwise noted, you simply will not be able to comprehend the Trinity - we are unable to comprehend the incomprehensible God. In A.W. Tozer's book "Knowledge of the Holy" (pages 7-8) he describes this problem thusly:
The effort of inspired men to express the ineffable has placed a great strain upon both thought and language in the Holy Scriptures. These being often a revelation of a world above nature, and the minds for which they were written being a part of nature, the writers are compelled to use a great many “like” words to make themselves understood.
When the Spirit would acquaint us with something that lies beyond the field of our knowledge, He tells us that this thing is like something we already know, but He is always careful to phrase His description so as to save us from slavish literalism. For example, when the prophet Ezekiel saw heaven opened and beheld visions of God, he found himself looking at that which he had no language to describe. What he was seeing was wholly different from anything he had ever known before, so he fell back upon the language of resemblance. “As for the likeness of the living creatures, their appearance was like burning coals of fire.” The nearer he approaches to the burning throne the less sure his words become: “And above the firmament that was over their heads was the likeness of a throne, as the
appearance of a sapphire stone: and upon the likeness of the throne was the likeness as the appearance of a man above upon it. And I saw as the colour of amber, as the appearance of fire round about within it.... This was the appearance of the likeness of the glory of the Lord.”
Strange as this language is, it still does not create the impression of unreality. One gathers that the whole scene is very real but entirely alien to anything men know on
earth. So, in order to convey an idea of what he sees, the prophet must employ such words as “likeness,” “appearance,” “as it were,” and “the likeness of the appearance.”
Even the throne becomes “the appearance of a throne” and He that sits upon it, though like a man, is so unlike one that He can be described only as “the likeness of the appearance of a man.”
When we try to imagine what God is like we must of necessity use that-which-is-not God as the raw material for our minds to work on; hence whatever we visualize God to be, He is not, for we have constructed our image out of that which He has made and what He has made is not God. If we insist upon trying to imagine Him, we end with an idol, made not with hands but with thoughts; and an idol of the mind is as offensive to God as an idol of the hand.
”The intellect knoweth that it is ignorant of Thee,” said Nicholas of Cusa, “because it knoweth Thou canst not be known, unless the unknowable could be known, and the invisible beheld, and the inaccessible attained.”
”If anyone should set forth any concept by which Thou canst be conceived,” says Nicholas again, “I know that that concept is not a concept of Thee, for every concept is ended in the wall of Paradise.... So too, if any were to tell of the understanding of Thee, wishing to supply a means whereby Thou mightest be understood, this man is yet far
from Thee.... forasmuch as Thou art absolute above all the concepts which any man can frame.”
Left to ourselves we tend immediately to reduce God to manageable terms. We want to get Him where we can use Him, or at least know where He is when we need Him. We want a God we can in some measure control. We need the feeling of security that comes from knowing what God is like, and what He is like is of course a composite of all the
religious pictures we have seen, all the best people we have known or heard about, and all the sublime ideas we have entertained.
So that being said, my notes:
"the Trinity is not one entity with three different aspects"
Correct; This is a heresy know as Modalism or Sabellianism
"But rather three entities with different personalities with a shared essence and relationship with each other"
Incorrect; this is a heresy known as Arianism which was expressly spoke against at the First Council of Nicea
"the Son, the Father, and the Holy Ghost wouldn't be ONE God, would they?"
Yes, they/he would be. This is a paradox. God is both one God and 3 persons (not to be confused with IN 3 persons - as if God could be divided) who shares a single divine will and yet has 3 different and distinct wills. Similarly it is not accurate to say that God is 3 persons or that God is one. It is only accurate to say both.
So, now to your analogies:
While you and your friends may share the attribute of being human this is not the same as being "of the same substance" or ὁμοούσιος (Homooúsios). (in fact, being created in the image of God, we share attributes with God) The same is true in all of your analogies. All of these similar objects are ὁμοιούσιος (Homoioúsios) or of a similar substance. In fact, it may be said that only the members of the Godhead are ὁμοούσιος (Homooúsios) - a new classification that was developed to describe God. It may be helpful to explore the origins of this idea - these ideas had been percolating for some time before Christ arived and these ideas were subjects of discussion by the ancient Greek philosophers.. It may therefore be helpful to read an overview of their discussion so as to contextualize the terminology used that often sounds strange to modern ears and understand why the concept of οὐσία or "substance" was adopted for the purposes of discussion by early Christian thinkers. In your analogy of the Greek gods, they are Distinct from the concept of the Trinity in that each God is clearly seperate from the next - the Triune Yahweh is indivisible and Jesus and the Holy are both seperate from God and yet the boundaries and distinction between them are not. These three repsent one single indistinguishable entity with no clear seperation between the three members of the Godhead.
Finally, to address your opinion that Christians are trying too hard to market this as monotheism, I would encourage you to take this idea seriously and consider it fully. As explained here, while one has salvation upon "believing in" Jesus, very quickly the question of who you actually "believe" Jesus to be very becomes important. If you do not believe who he said he was (eg, his statements about the Trinity, then you do not actually believe in Jesus, but something else.
So in summary,
What is it that makes them out to be one? in which sense are they one?
They are one in the sense that they are ὁμοούσιος (Homooúsios) - something for which the translation "same-substance" is not quite an accurate as we don't quite have a modern concept or term for (outside of "triune" that is).