But if I want to match the results that the website gets for "AABBCCDD" as HEX input (https://crccalc.com/?crc=AABBCCDD&method=CRC-16/X-25&datatype=hex&outtype=0), which gives 0xF486, I use $cont=0 (I believe the cont parameter is "continue", and since you aren't adding to the string, but using it as the whole message, you want cont=>0 ) and $input = "\xAA\xBB\xCC\xDD" and it gives the same 0x4FCB that the online tool gives.

Fixing your cont->$cont typo to cont=>$cont , and adding a few different test cases, the test_it function has the right setup to get the same results as the online tool, as shown by the Test::More results.

#!perl.exe

use 5.014; # //, strict, say, s///r
use warnings;

use Digest::CRC;

my ($crc, $input, $width, $init, $xorout, $refout, $poly, $refin, $con
+t, $digest, $ctx, $exp);

$width  = 16;
$init   = 0xffff;
$xorout = 0xffff;
$refout = 1;
$poly   = 0x1021;
$refin  = 1;
$cont   = 0;      # ???

$input = 0xaabbccdd;

$ctx = Digest::CRC->new(width=>$width, init=>$init, xorout=>$xorout,
         refout=>$refout, poly=>$poly, refin=>$refin, cont=>$cont);
$ctx->add($input);

$digest = $ctx->hexdigest;

say $digest;


use Test::More;
use feature 'fc';
use Data::Dump qw/pp/;
sub test_it {
    my($in, $cont, $expect) = @_;
    my $ctx = Digest::CRC->new(width=>$width, init=>$init, xorout=>$xo
+rout,
         refout=>$refout, poly=>$poly, refin=>$refin, cont=>$cont);
    $ctx->add($in);
    my $got = $ctx->hexdigest;
    is fc($got), fc($expect), sprintf("test_it(%s, %s, %s)", pp($in), 
+$cont, $expect);
}
test_it("AABBCCDD", 0, '1D8D'); # https://crccalc.com/?crc=AABBCCDD&me
+thod=CRC-16/X-25&datatype=ascii&outtype=0
test_it("\xAA", 0, 'FA28');     # https://crccalc.com/?crc=AA&method=C
+RC-16/X-25&datatype=hex&outtype=0
test_it("\xAA\xBB\xCC\xDD", 0, '4FCB');  # https://crccalc.com/?crc=AA
+BBCCDD&method=CRC-16/X-25&datatype=hex&outtype=0
test_it("\xDD\xCC\xBB\xAA", 0, 'F486');  # https://crccalc.com/?crc=DD
+CCBBAA&method=CRC-16/X-25&datatype=hex&outtype=0
done_testing;
[download]

__END__
1680
ok 1 - test_it("AABBCCDD", 0, 1D8D)
ok 2 - test_it("\xAA", 0, FA28)
ok 3 - test_it("\xAA\xBB\xCC\xDD", 0, 4FCB)
ok 4 - test_it("\xDD\xCC\xBB\xAA", 0, F486)
1..4
[download]

I would have expected either test3 or test4 to match your 0xAABBCCDD input, but neither did.

And then I realized: 0xAABBCCDD is the integer 2864434397, which Perl stringifies into "2864434397", so if I try that string input, the tool gives 0x1680, which is what Digest::CRC gives for your $input = 0xaabbccdd; input.

Digest::CRC is built to work on a series of octets (because that's what it gets from reading a file), so when you pass in a number, it lets Perl stringify it into a string, which it treats as a series of octets; it doesn't deal with the raw octets for the integer (that would open the big-/little-/middle-endian can of worms). (Presumably, you could pack your integer as well, but that is left as an exercise for the reader.)