The best way to explain the trinity is with an example. Picture a man, name Joe, who is a husband to his wife, a father to his children, and a child to his parents. He is one person, playing different roles, depending on who is dealing with. This is not a multiple personality disorder. Rather this is a way for a person to optimize his influence on his family. Polytheism was more like a multiple personalty disorder.
Joe is intimate with his wife in ways he is not with his children and parents. He is respectful of his parents, in ways that differ for his wife and children. He is firm but fair with his children, in ways unique to them. He is one person with three hats. This is different from a multiple personality disorder; polytheism. In that case, each hat becomes a unique person instead of the same person specialized to needs.
God the father is how God relates to the Jews and Arabs. God the son is how God relatives to the Christians While God the Holy Spirit is how God relates to those with the charism of faith in the Holy Spirit.
An interesting scenario is holiday time when all the family gets to together. In this case, Joe has to deal with all this his family at the same time, and often together. His parents may want spend time with his children, spoiling them. Joe has learn how to be respectful of this parents, even when they will make it harder for him to be firm but fair with his children. He does not remain with only the three original masks. Or his wife and mother may both wish to control Joe, so Joe has to walk the line and not alienate either. This is where the father and son, or the son and Holy Spirit, or even the father, son and holy spirit, may all have to merge into one or two to optimize the day.
In polytheism, each god is specialized and autonomous all by itself. Aphrodite was a unique person who specialized in love and desire. Zeus is separate and distinct. The trinity is one person with three masks. Someone with a multiple personality disorder would have more than one autonomous personality, that may not be on the same page as the rest of the characters. The trinity is alway on the same page in terms of the needs of the family.