Orion syncs history, natively though iCloud sync. Battery consumption should be on par or better than Safari (because Orion has native ad-clocking). If this isn't the case, it is unusual and worth digging deeper into.
Hey, Vlad! History did not sync between Orion on my MacBook and Orion on my iPhone. Bookmarks synced, however. I was running the latest version on all devices. Perhaps it's PEBKAC; I'll give it another go, as using Orion was an excellent experience otherwise.