But where does that key get set?

There's init_MAKE(), which sets it (if it's not alredy set at this point) from the environment, or else from $Config{make} (which should be what you get via perl -V:make )

sub init_MAKE {
    my $self = shift;

    $self->{MAKE} ||= $ENV{MAKE} || $Config{make};
}
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A couple of well-placed prints can answer many questions, e.g. allow you to check whether the value is what you'd expect, or something funky

sub init_MAKE {
    my $self = shift;

    print STDERR ">>> before: '$self->{MAKE}'\n";
    $self->{MAKE} ||= $ENV{MAKE} || $Config{make};
    print STDERR ">>> after:  '$self->{MAKE}'\n";
}
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(and if you don't get any print output at all, you'll at least know you're executing something else entirely...)

P.S. For a discussion of !!, see Perl Idioms Explained - !!expr and Secret Perl Operators: the boolean list squash operator, x!! (and probably others) — if generally interested, it's not really relevant to the problem at hand.