I don't think we are given explanations by D&D. They didn't expand on the lore of the Ice Dragon, either in the show, or in interviews afterwards. At least to my knowledge, I kept pretty up-to-date during Season 8, but I could have missed it.
If you want George's take, I would suggest reading his novella ["The Ice Dragon"](https://www.amazon.com/Ice-Dragon-George-R-Martin/dp/0765378779)
Now my two cents - I think the Viserion's blue flames were not 'fire' but intense cold. If you have seen videos involving liquid Nitrogen, you will be familiar just how destructive it can be. Physical matter gets so brittle, it just crumbles into a pile of dust. If you pour liquid Nitrogen on ice, it gets so cold, it just crumbles if you smash it. My guess is the portion of the Wall the Ice Dragon's 'flames' hit contracted immensely to such an extent, it just broke the ice.
If the Wight Dragon is anything like the Ice Dragon in his previous books, it will breathe frost.
You assume the White Walkers don't also have dragons...
Edit: This is not ASOIAF book, but clearly the idea is there...
> It is unknown whether the Ice Dragon had ever existed in the Song of Ice and Fire world
It is known: https://www.amazon.com/Ice-Dragon-George-R-Martin/dp/0765378779