I think that’d go hand in hand with aggressive* monotheism : upon the consolidation of manygods into monogod, the one god naturally had to take the powers of all the minions, and even be more powerful than the pantheon combined. Otherwise the religious pissing contest would be all skewed in favor of pantheism : they could just zerg the monogod, or something.
So you start with a single Gawd that knows more, can do more, and is less of a random evil jerkwad than the older gods of the neighbouring tribes. Considering the hijinks of the Roman gods, the latter is a far cry from omnibenevolence, but still ;).
From that point on, whenever the monotheistic group encounters another religious group, be they pantheists or monotheists as well, the Monogod can’t but take from them whatever attributes He doesn’t have yet - if He’s the one true God, He must beat all the other gods, He can’t be the one true God if some other god can, I dunno, stop the Sun dead in its tracks and He can’t.
Since Buddha is omnibenevolent, and Odin is omniscient (or nigh-omniscient ? I must confess my Norse lore ain’t what it used to be. Skald, an expert counsel if you please ?), and there’s a god out there for every single power and every natural phenomena, well… overtime, you’re bound to wind up with an omni* God.
So, IMO, the omni* tradition comes from missionary PR, simple as that. You can’t convert anyone to your faith if your God ain’t better than theirs.
*not in the sense of being warlike, mind you, but in the sense of aiming to convert other people to the creed - a Christian novelty.