No, caller in the logMessage function itself does something different. Witness:

{
    package My::Logger;
    sub import {
        my $PACKAGE = caller;
        no strict 'refs';
        *{"$PACKAGE\::log_message"} = sub {
            my ($message) = @_;
            print STDERR "$PACKAGE - $message";
            };
    }
}
{
    package My::Utils;
    My::Logger->import;
}
{
    package main;
    # outputs "My::Utils - Hello", not "main - Hello"
    My::Utils::log_message("Hello");
}
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The log_message function outputs what package it has been installled into, not called from.

While this is rarely useful, it does occasionally have uses. Object::Stash uses something along these lines.