You can count the occurrences of each name with a hash and use eval substitution to increment on the go. Like this:

use strict;
use warnings;

my %count;
while (<>) {
    s{<bib id="CIT-(\w+)-b(0*\d+)">}
    {"<bib id=\"CIT-$1" . ++$count{$1} . "-b$2\">"}ge;
    print;
}
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Incrementing undef will treat it as 0, so when a key is first encountered it will have 1 appended to the name.