Here is Wikipedia's description of an argument from analogy:
Argument from analogy is a special type of inductive argument, whereby perceived similarities are used as a basis to infer some further similarity that has yet to be observed. Analogical reasoning is one of the most common methods by which human beings attempt to understand the world and make decisions. When a person has a bad experience with a product and decides not to buy anything further from the producer, this is often a case of analogical reasoning. It is also implicit in much of science; for instance, experiments on laboratory rats typically proceed on the basis that some physiological similarities between rats and humans entails some further similarity (e.g. possible reactions to a drug).
One can also find counter-arguments using analogy:
Arguments from analogy may be attacked by use of disanalogy, counteranalogy, and by pointing out unintended consequences of an analogy.
Paul Bartha notes:
Analogies are widely recognized as playing an important heuristic role, as aids to discovery.
Arguments from analogy or counteranalogy are heuristic aids and potentially persuasive. They are not deductive arguments and so they are imprecise.
Keeping this use of analogy in mind one can perhaps more fruitfully look for arguments for or against the idea of the Christian Trinity by seeking out those presenting analogies for or against the idea.
Not all those associated with Christianity accept the Trinity. Here is a presentation of analogy and counteranalogy given by Ray Pritchard with respect to the Jehovah's Witnesses:
Sometimes the Jehovah's Witnesses (who pointedly deny the Trinity) ridicule it with this little equation: 1 + 1 + 1 = 3. In their minds Christians worship three Gods, not one. The answer is quite simple. The doctrine of the Trinity is not absurd if that's what the Bible teaches. Furthermore, there is more than one way to play with equations. You could also say it this way: 1 x 1 x 1 = 1!
Note the counteranalogy that belief in the Trinity is analogous to "1 + 1 + 1 = 3". Note the analogy countering the counteranalogy that rather belief in the Trinity is like "1 x 1 x 1 = 1".
For Pritchard, ultimately, it is a matter of faith. For those looking for a deeper understanding the use of analogy (and counteranalogy) can help someone decide whether they will want to believe in the Trinity or not.
For an introduction to analogy from a philosophical perspective, see Paul Bartha's "Analogy and Analogical Reasoning".
Bartha, Paul, "Analogy and Analogical Reasoning", The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2016/entries/reasoning-analogy/.
Pritchard, R, "God in Three Persons a Doctrine We Barely Understand", Christianity.com https://www.christianity.com/god/trinity/god-in-three-persons-a-doctrine-we-barely-understand-11634405.html
Wikipedia, "Argument from Analogy" https://en.wikipedia.org/wiki/Argument_from_analogy