I think that “pronomono” has the best logical explanation, saying that “they are one in the sense they are working in a unified effort toward a common goal.”
The term “trio” (or trinity) can be used for a musical performance by three individuals, not some physically doubly-conjoined-since-birth scenario. As far as the Holy Trinity goes, I would think that the intended essence of that doctrine is that of three separate, individual entities which each manifest a distinctly different type of manifestation, aspect, or “mode” of the Divine God – including God Himself, who is the Almighty Creator; as well as his chosen human representative, his “son,” Jesus Christ; and his active force, his holy spirit – all three of which being physically, or substantially, separate, but all three such fundamental elements being united in an embodiment and manifestation of “the divine.”
Think of it this way: A composite of the fundamental elements, or manifestations, or “modes,” of a government could be the country’s president, a top military general, and a Tomahawk cruise missile system. The president is the fundamental element of the federal government administration, the top military general is the representative in the flesh you might see in a military operation, and the Tomahawk cruise missile system – well that is the actual tool used to physically administer the will of the government (the thing that manifests the power to go right up your ass and blow your body’s cells umpteen ways from Sunday if you’re one of those ISIS folks). All three elements (president, general, missile) are a trio, or trinity, of whoop-ass that gets the job done. Simple.