That is because the i/o system and the error handling behave in different ways. For the error handling there is a central server process registered under the name :standard_error but this is not case for the i/o system. The best way to view an erlang/elixir system to think OS with many tasks and each one has its own i/o system.
You will find a description of this in the link in the erlang rationale I wrote many years ago Robert on anything: The Erlang Rationale . Towards the end it describes the i/o system and the “task” structure in erlang. You actually hit when you do a ^G in the shell.
Strangely enough this was the first time much of the rationale for erlang was actually described, about 10-15 years after it was designed. This was because we had so thought about it and discussed it we felt it was self-evident that this was the way it should be done. We found out later that not everyone thought the same way, some had not grasped the beauty of our design. 
In The Erlang I/O Protocol — OTP 29.0.2 (stdlib 8.0.1) you will find a description of the i/o-protocol as it is today.
