Since I associate the
Trinity with God, I find it too much for name. Perhaps people who don't mind using
Jesus or
Emmanuel as names would be fine with
Trinity, but I personally would feel uncomfortable.
For whatever reason, some people who use the name
Trinity strike me as less religious and more quasi-spiritual (for lack of a better term). It is a bit like the use of Bodhi. A lot of the people who use Bodhi as a name are not devote Buddhists, but they like the association with enlightenment and spiritual awakening.