The name hypogeum comes from the Greek hypógeion, consisting of hypo (under) and gẽ (earth). It was adapted to hypogaeum in Latin. The term is used for pagan cult sites, specifically temples or tombs. However, only for sites from the Stone Age to Roman times.
We do not use this term, we just want to explain it because it appears in proper names. It is quite common in the Mediterranean region, for example Malta or southern Italy. From the Greek origin of the word, it is actually a generic term for something underground, it is simply the Greek translation of the Latin term Subterranea, which we use. However, the usage is completely different, it is only used to describe religious structure, temples and tombs of great antiquity. The time span ranges from the Stone Age to Roman times. They are exclusively man-made structures and exclusively non-Christian. Christian churches are called cave churches, Christian graves are called catacombs.
But the term hypogeum is unfortunately quite misleading, as it is also used in literature in a Christian context. Moreover, pagan sites are also referred to as catacombs. In other words, the term is so vague and is used so arbitrarily that it is not suitable for classifying subterranea. And besides, it is not really relevant to distinguish the Christian religion from all others, except perhaps for fanatical Christians.