My wife and I walked Tongariro (actually, the whole circuit, not just the crossing) many years ago on our honeymoon. That was real backpacking through wilderness (even if we did not need to carry stoves and tents because of the excellent shelters).
What you will find similar is the comradery of the mountain hut, especially after doing the crossing in a blinding rainstorm, as we did (see photo). But that is about it. The Camino is, with very few exceptions, not in natural areas--it is in agricultural areas. The Camino does not seek to go through the most scenic areas--it follows the lowest, easiest route. The Tongariro Crossing is a one-day sprint; the Camino is a marathon.
On some level, by biggest misconception about the Camino was that it was just a long backpacking trip. The Camino is not really backpacking in any sense except carrying a backpack. Much of the Camino is not on trails, but on roads, gravel if you are lucky, but very often concrete sidewalks or asphalt. The Camino is routed through villages and hamlets, not away from them through nature. And by far my biggest failure was to under-appreciate just how much longer the Camino was, and how that changed how I had to care for my feet. Just padding blisters with molefoam would not work. I had to prevent them from forming in the first place, which meant changing socks every 90inutes (summer Camino), foot powder, no hiking boots (the Camino is not rough enough to need real hiking boots), but Keen sandals and well-ventilated trail shoes or runners designed for concrete/asphalt.
Buen Camino