I would read each record complete rather than one line at a time, and to keep it simple, would push scores into an anonymous array per student. Then just iterate over each student and do the math.

use List::Util qw( sum );

my %student;

local $/ = '';

while( <DATA> ) {
  push @{$student{$1}}, $2 and next if m/\A([^\n]+).+?(\d+)\n?\Z/s;
  warn "Invalid record #$.: <<$_>>\n";
}

while( my( $name, $scores ) = each %student ) {
  print "$name: Average ", sum( @{$scores} ) / @$scores, "\n";
}

__DATA__
Jack
Student ID - 12445
Math Score - 45

Jill
Student ID - 234254
Math Score - 90

Jack
Student ID -12445
Math Score2 - 33

Jill 
Student ID - 234254
Math Score2 - 10
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