I would encourage passing in via \*STDOUT over *STDOUT. Globs are a type from Perl 4 and look too much like strings. Checking !ref($arg) is much more common than checking "GLOB" eq ref(\$arg) and !ref($arg) needs to be checked anyway in order to deal with lexical file handles and \*STDOUT and *STDOUT{IO}.

Supporting passing in "STDOUT" gets rather tricky. Consider:

package Whatever;

use My::Logger qw< setLogLevel >;

setLogLevel( STDOUT => 1 );
open LOG, '>>', 'some.log' or die;
setLogLevel( LOG => 2 );
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My::Logger::setLogLevel() needs to treat 'STDOUT' as *STDOUT (or *main::STDOUT) but must treat 'LOG' as *Whatever::LOG.

    my( $arg )= @_;
    my $fh;
    if(  ref( $arg )  ) {
        $fh= $arg;
    } elsif(  "GLOB" eq $arg  ) {
        $fh= \$arg;
        # or  $fh= *{$arg}{IO};
    } else {
        # It is a plain string.  If you want to
        # treat it as the name of a glob, then
        # you'll need to check for /::|'/ and
        # prepend caller().'::', unless it is
        # special like 'STDOUT'.
        # You'll also need 'no strict;' here.
        ...
    }
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