Is there a way to write a Bash function which aborts the whole execution, no matter how it is called?

What you could do, is register the top level shell for the TERM signal to exit, and then send a TERM to the top level shell:

#!/bin/bash
trap "exit 1" TERM
export TOP_PID=$$

function func()
{
   echo "Goodbye"
   kill -s TERM $TOP_PID
}

echo "Function call will abort"
echo $(func)
echo "This will never be printed"

So, your function sends a TERM signal back to the top level shell, which is caught and handled using the provided command, in this case, "exit 1".

You can use set -e which exits if a command exits with a non-zero status:

set -e 
func
set +e

Or grab the return value:

(func) || exit $?

A child process can't force the parent process to close implicitly. You need to use some kind of signaling mechanism. Options might include a special return value, or perhaps sending some signal with kill, something like

function child() {
    local parent_pid="$1"
    local other="$2"
    ...
    if [[ $failed ]]; then
        kill -QUIT "$parent_pid"
    fi
}