If the text within your OP represents a single line from a closed log file, i.e., if you are not trying to tail a constantly growing log file, the following might represent a possible approach to a solution (untested):

use warnings;
use strict;

my $log_file  = 'file.log';
my $uid       = 'xy1234';
my $threshold = 20;
my @occurrences;

open my $fh_log, '<', $log_file or die "opening $log_file: $!";

LINE:
while (defined(my $line = <$fh_log>)) {
  next LINE unless $line =~ m{ uid=$uid }xms;
  push @occurrences, $line;  
  }

close $fh_log or die "closing $log_file: $!";

if (@occurrences > $threshold) {
  print "occurrences for $uid above $threshold \n";
  print @occurrences;
  }
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Of course, the  print statements at the very end will have to be replaced with code to compose and send an appropriate e-mail message, etc., but I assume you know how to do that. (In any event, I'm not the best one to ask about that.)