Sometimes when you select appear offline, you are still online and others can still initiate whispers to you or see you in game. But someones being online/idle/busy, you are appeared as offline to other players.
Depending on the game that is not always the case. The Bnet launcher ensures you don’t show up in that chat service. However, if you play a game that has other social or chat features you will still show up in that.
Example - you are offline in the Bnet launcher, but in WoW Guild you will show up as online. Same with anyone you added as individual characters to the in-game WoW chat system.
Here is the 2017 News article announcing it and explaining it is not applicable to every in-game function.