This isn't what I would consider a golfy solution, but it is more brief:

my @data;
for my $file (qw(this.txt that.txt)) {
  open my $fh, '<', $file or die $!;
  push @data, [];
  while( <$fh> ) {
    push @{$data[-1]}, $_ if /\n\z/ && ! /^>/;
  }
}
[download]

If you want that in the form of a subroutine:

@data = read_files( qw(this.txt that.txt) );

sub read_files {
  my @data;
  for my $file ( @_ ) {
  open my $fh, '<', $file or die $!;
  push @data, [];
  while( <$fh> ) {
    push @{$data[-1]}, $_ if /\n\z/ && ! /^>/;
  }
  return @data;
}
[download]

(Untested)