I actually have that kind of experiences (mostly on a subtle level, but sometimes more intense, to the point that tears might well up) regularly as a result of my meditation/spiritual practice. For me it is more "seeing the divine expressing itself through the person" or feeling a particular kind of universal personhood in a tree, as if that particular tree stood outside of time, representing all of life. It can also happen with my own emotions: sometimes I will feel sad or angry, and the feeling will transform from "my anger" to "an expression of cosmic anger" or "the Divine expressing itself through me as Anger".

It might be a bit different from the experience described in the citation, but to me that description definitely sounds familiar (in particular the simultaneity of the particular and the universal). The effect of my practice on my daily life and the way I relate to it also sound pretty similar to the way some use psychedelics for spiritual or therapeutical practice. I actually lost quite a lot of my appetite to experiment with psychedelics since I started this kind of practice.

I mostly practice following the teachings of Rob Burbea, but I am also fond of reading texts from mystics of the Abrahamic religions, and this kind of experience is also reported regularly in that context. Mother Theresa for instance, when asked how she could stand being at the contact of so much suffering, said something along the lines of "I look into the eyes of my patient, and what I see is the glaze of my beloved Christ".