I have come across various viewpoints, but they mostly fall into two camps:
1. Jesus IS NOT identical to God but is part of the Godhead in the way that a human son is part of his father. They share genes plus physical, emotional and mental characteristics, but they think and act independently.
2. Jesus IS God, i.e. he is fully one and the same ‘being’ as God and the reason there are separate names is to distinguish the part of God the father who remained in Heaven while his alter ego Jesus the Son was on earth. In this case, it appears that God decided to split himself into two different 'beings' by begetting a son. He can thereby appear in two places at once and his two personas can act autonomously to each other, but remain the same entity.
Another analogy for the first scenario could be a band comprising a guitarist, a bassist and a drummer. The word ‘band’ is a singular item grammatically although in this case it’s also a trio. We think of it as one thing, however its constituent members are separate independent beings. The band members each make a different sound, but to make music, they have to be in harmony with each other and keep to the same rhythm and timing.
It's much harder to find an earthly analogy/metaphor for the second scenario, especially bearing in mind Jesus allegedly existed before the universe was created. Conjoined twins later separated doesn’t work as there are two distinct individuals involved with that. Nor does genetic cloning fit as the offspring is not ‘as one’ with the parent, he/she is merely identical genetically. I'd be interested to know if anyone has any suggestions that are not sci-fi or mythology.)
In either scenario, people tend to only focus on Jesus but what about the third part of the Trinity, the Holy Spirit? Is it simply God’s consciousness, his brain and thoughts and wishes, or is he/she/it a separate but closely linked entity which God controls and sends to do his bidding?
If the former, why not just make the godhead a duality, why a trinity? If the latter, why not give the Holy Spirit a personal name?
Or could it be that humans invented God and later added the idea of ‘three in one’ following other earlier myths, traditions and superstitions? Trinity appears to imbue the deity with greater majesty, power and mysticism and the concept is great PR - “buy one, get two free”!