Is it possible to configure the terminal in OS X to show
stderr output automatically in a different colour than
stdout output from a process?
Or is this only possible if the process is outputting coloured output?
There is no straightforward solution, but there are at least a few workarounds available to do this.
You can use the command from this answer on serverfault (substitute
command with the name of your script/command) to make stderr appear in red (for other colors, refer the ANSI escape sequences list and change the 31m in the below command accordingly):
command 2> >(while read line; do echo -e "\e[01;31m$line\e[0m" >&2; done)
You can use stderred to color stderr in red.
You may want to have a look at stderred which allows to output Stderr in red.
It's OSX compatible and there is a manual to install it on OSX.