The error is not passed as an argument to your die handler, but is in $@.

How so? $@ is the last eval error. The arguments to die are passed to the DIE handler (whether invoked by perl itself or via die):

local $SIG{__DIE__} = sub {
    open my $fh, '>>', 'log.err'
        or warn "Can't append to 'log.err': $!" and return;
    print $fh @_, "\n\$\@: |$@|";
};
eval { die "fubar!"; }; # <--- update, semicolon was missing
1 / 0;
__END__
Illegal division by zero at foo.pl line 8.
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Contents of log.err:

@_: |fubar! at foo.pl line 7.
|
$@: ||
@_: |Illegal division by zero at foo.pl line 8.
|
$@: |fubar! at foo.pl line 7.
|
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The fubar! at foo.pl line 7. is repeated since $@ hasn't been cleared. If there had been a subsequent succeeding eval, $@ had been cleared.

Neither die messages nor system errors ($!) are shoehorned into $@ for the DIE handler.

update 2:

Also, your __DIE__ handler will also be invoked from within eval { ... } blocks, which is most likely only confusing to you. You should check $^S to see if eval is currently running:

    return if $^S; # we are within an eval block
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This, as a general advice, is misleading. One might very well be interested in eval errors.