This is a partial answer hopefully getting to another question that deals with the apparent multiplicity in God that the Trinity represents.
Not all religions associated with Christianity need to assert the idea of the Trinity. For example Unitarian Universalists with roots in liberal Christianity do not appear to do so:
Unitarian Universalism (UU) is a liberal religion characterized by a "free and responsible search for truth and meaning". Unitarian Universalists assert no creed, but instead are unified by their shared search for spiritual growth. As such, their congregations include many atheists, agnostics, and theists within their membership. The roots of Unitarian Universalism lie in liberal Christianity, specifically Unitarianism and universalism. Unitarian Universalists state that from these traditions comes a deep regard for intellectual freedom and inclusive love. Congregations and members seek inspiration and derive insight from all major world religions.
What is important in the argument is to identify more precisely who holds these beliefs.
Once one knows who holds these beliefs, one needs to find representatives from those groups, preferably the most respected representatives, and quote their exact views on the Trinity paying attention to their arguments.
Then one can attempt to use those arguments to try to reach a contradiction.
Without getting those quotes and identifying whom one is opposing, one is setting up a straw man. This is how Wikipedia describes a straw man:
A straw man is a common form of argument and is an informal fallacy based on giving the impression of refuting an opponent's argument, while actually refuting an argument that was not presented by that opponent. One who engages in this fallacy is said to be "attacking a straw man."
The issue of having multiplicity in unity in general should raise philosophical concerns which are worth exploring even with particular examples such as the three Persons in one God in the concept of the Christian Trinity. One simply needs to be precise about what people are claiming when they speak of these particular multiplicities.
Wikipedia, "Straw man argument"
Wikipedia, "Unitarian Universalism"