Did a Google. Starcraft 2 replay files are only a few hundred KB each. I'd wager they have A LOT more to keep track of there.
Back to swgoh, you don't store everything you described. You don't need to know the mods. You just need the rng seed, base stats, ability levels, and a move/target log. The game engine can recreate the outcomes upon request.
Yes, you fo need to check the log vs results. They already store the end state results. That alone may be enough to compare against. But they could easily add a battle state snapshot every 10 or 20 moves as well, I suppose.
Yes, this is not zero data to store. But, data storage is very cheap these days. The technical limitations are nonexistent. The only reason we don't have a rudimentary system in place is because it won't make CG money.