As explained by Lotus1 &stat() isn't the same thing as stat(). &stat actually is the function in the *main::stat glob, which by default does not exist:

use v5.14;
use strict;
use warnings;

open my $fh, '>', "tmp_1222046.txt";
say 'stat($fh): ', join " ", stat($fh);

say '*::stat{CODE} is ', *::stat{CODE} // 'undef';

say '$sub = \&stat;';
my $sub = \&stat;

say '$sub: ', $sub;
say '*::stat{CODE} is ', *::stat{CODE} // 'undef';

say 'stat($fh): ', join " ", stat($fh);
say '$sub->($fh): ', join " ", $sub->($fh);
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stat($fh): 0 0 33206 1 0 0 0 0 1536590875 1536590875 1536590555
*::stat{CODE} is undef
$sub = \&stat;
$sub: CODE(0x340be0)
*::stat{CODE} is CODE(0x340be0)
stat($fh): 0 0 33206 1 0 0 0 0 1536590875 1536590875 1536590555
Undefined subroutine &main::stat called line 18.
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So accessing \&stat actually creates an entry in the symbols table, which is different from the function called with stat()