I wonder if part of the problem is the actual statement of the doctrine of the Trinity you have heard, because there are certainly versions I do not agree with either, and others which are declared anathema and heretical.
correct statement of the doctrine:
The Father, Son, and Holy Spirit are three distinct persons but only one God.
statements of the doctrine which go too far and I do not agree with:
God is three persons.
God is triune.
statements of the doctrine which are heretical:
God is made up of three persons.
Father, Son, and Holy Spirit are the three parts of God.
God has different aspect or modes known as the Father, Son, Holy Spirit.