And so the conflict between “religion” and “spirituality” wears on through millenia.
A lot of, perhaps even most people, do not have an inherently spiritual and holistic view of life. Even people who ascribe to some kind of religion often separate that as a thing unto itself, "oh, that's religion, that's a separate thing, that's something I only deal with at church/temple/the mosque." Spirituality is different from religion and it's something that, eventually, some paths show us IS life. Tango, yoga, bodywork, the healing arts, art, martial arts, music, philosophy, advanced physics and math...they all live in the place where spirituality, working with the spirit, working with energy, whatever...IS life.
So, is tango religious? In the sense that religion is a separate and discrete thing that happens when you go to a special place and wear special clothes and hear special music and make special body shapes and are led by somebody and maybe you aren't thinking of God at all while you're there...then, my answer is, "maybe! It sure quacks like a duck!"
I've heard people call their milonga their “church.” Makes sense to this member of the congregation. But when we have a milonga in an actual church, Argentinians find it unsettling. Disturbing. Wrong.
I've heard people call tandas their “prayers.” Sure. Particularly if it's a good tanda, what else could it possibly be? Dance with me and know the face of God, well, sure, why not? Get closer to the essence of life, understand humanity and samsara and dharma a little better, feel something greater than ourselves, struggle, catch a glimpse of something divine...definitely sounds like the motivations behind a religious practice.
So, is tango spiritual?
I'll say yet again: it reflects its practitioners. If you're doing it from an exclusively physical standpoint, wanting nothing else, then it's exclusively physical. The more spiritual YOU are, the more spiritual IT is.
I'm afraid I say the same thing about sex.
By the way I hate that word, "spiritual." (Much the way Alejandro hates the word "organic"--words that have been so laden with new-age plaque that their objective meanings have been lost.) I only use it because we don't have another one that means of or pertaining to the inner and or outer spirit or spirits. If by spirit you mean the divine spark that makes life go.
Is tango against your religion?
If you say it is, then you have not yet begun to understand the first thing about any religion on Earth.