For PlayStation users, the most effective solution reported by the community is downloading anything else from the PlayStation store first. This can include free PS Plus games or a game from your library that you haven't played before. Strangely enough, this seems to refresh the Diablo 4 license, with many confirmations for this method on Reddit.
Xbox players are experiencing the error due to an Early Access game sharing issue, so they will need to rely on Diablo 4 local couch co-op for now. Blizzard supports cross-progression systems for Diablo 4, with both characters saved on the respective Battle.net accounts. This restriction should be lifted on June 6 alongside the worldwide launch.
PC players encountering the same error code should select Restart and Update from the top-left down-down menu in the Battle.net launcher. Thankfully, the issue seems to be mostly fixed here.