Seems weird. And what really smells suspicious is that if you switch the order of the alternatives, the second string suddenly matches the regex.

for my $regexp (qr/^((a*)b|a*b?d)*c\2$/, qr/^(a*b?d|(a*)b)*c\2$/) {
    for my $string (qw( aabadcaa aababdcaa )) {
        say $string =~ /$regexp/ ? 'Yes' : 'No';
    }
}
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Output:

Yes
No
Yes
Yes
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If you want to know what Perl's regex engine does, not just guess, run the script with

use re 'debug';
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