Your program logic is actually pretty good. Without more information on what the school name can be, I'd say the only way to determine whether there is a school specified -- if other non-school-address data is allowed after "/data_..." -- is so check your second regex capture against known-good or known-bad values.

Or, if it's always a US-based school web address, then checking that it ends in .edu (or I guess .net, .com, or .org, and maybe .us) might be enough.

As for readability and best practices, I'd write that snippet as follows, tested with the provided data:

#! /usr/bin/perl;
use strict;
use warnings;
use autodie; # errors from open made fatal

sub DEBUG { 1 }

my $filelist = 'tmp.txt';
open my $filelist_handle, '<', $filelist;
while (<$filelist_handle>) {
    chomp;
    my ($type, $school) = m!
      ^                 # anchor to beginning
      /home/test/       # common to all lines
      (\w{3})           # capture 'type'
      /\.date_[^.]+     # common to all lines
      (?:               # non-capturing group
        .+?(\w+)\.\w+$  # capture domain name?
        |               # or don't capture
      )                 # end group
    !x # /x flag means ignore white space in pattern
    or next; # skip line if it doesn't match
    
    # do extra check that $school is acceptable
    $school //= 'null'; # regex gives undef if not found
    
    if (DEBUG) {
        print "match: $_\n";
        print "\ttype: $type\n";
        print "\tschool: $school\n";
    } else {
        open my $line_handle, '<', $_;
        while (<$line_handle>) {
            print "Type:$type:School:$school:File:$_\n";
        }
    }
}
[download]

Example debug output:

match: /home/test/abc/.date_run_dir
        type: abc
        school: null
match: /home/test/def/.date_run_dir
        type: def
        school: null
match: /home/test/abc/.date_file_sent.email@wolverine.cole.edu
        type: abc
        school: cole
match: /home/test/abc/.date_file_sent.dp3.drew.net
        type: abc
        school: drew
match: /home/test/def/.date_file_sent.email@wolverine.cole.edu
        type: def
        school: cole
match: /home/test/def/.date_file_sent.dp3.drew.net
        type: def
        school: drew
[download]