Correction: God is One Being shared by three Persons. I understand what you are trying to say, but you need to distinguish between being (ontology) and person (identity).
The Father, the Son, and the Holy Spirit are the same ontological being. The distinction is in the persons that exhaustively share the one being of God. Only God can have His singular being shared by more than one person. We only have our being shared by one person. Those who are otherwise live in rubber rooms.
Thanks!
You are correct in this difference, as though subtle, not making this distinction can lead one to either errors of Modulism or tri trinity!