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cat $rrs.*.err.txt | grep "Error $err"

change it to

awk -v err="$err" '/Error/ && $0~err {for(i=1;i<=NF;i++)if($i~/ID/){print $(i+1);next}}' $rrs.*.err.txt

Another one:
awk '$0~e {if(match($0,/ID [0-9]+/)) print substr($0,RSTART+3,RLENGTH-3)}' e="Error $err" $rrs.*.err.txt
And the output will not have the colon generated by the previous solution.

Then the statements given to you should print one line of output for each matching line of input.