Here we go, implementing it revealed more ambiguities

use v5.12;
use warnings;
use Data::Dump;

my @in = do { local $/; eval <DATA> };

sub DBG { ddx @_ if 0 }                 # debug

DBG "INPUT"=> @in;

my @out;
my $similarity = 3;                     # number equal elements

my $last = shift @in;
my $sum  = [ @$last ];                  # init

while ( my $cur = shift @in ) {
    my @zeros = grep { $cur->[$_] == 0 } 0..1; 
    my @same  = grep { $cur->[$_] == $last->[$_] } 0..$similarity-1;

    if ( @zeros and @same == $similarity ) {
        my $non_zero = 1 - $zeros[0];

        DBG "SUM" => $last, $cur, \@zeros, $non_zero ;
        $sum->[$non_zero] += $cur->[$non_zero];
    }
    else {
        DBG  "OUT" => $last, $cur, \@zeros;
        push @out, $sum;
        $last = $cur;
        $sum  = [ @$last ];             # init
    }
}

push @out, $sum;

dd @out;

__DATA__
[ -1, 5, 1 ],
[ 0, 5, 1 ],
[ 0, 5, 1 ],
[ 1, 5, 1 ],
# separated repetitions
# [ 0, 5, 1 ],
# [ 0, 5, 1 ],
[ 3, 4, 1 ],
[ 5, 1, 1 ],
# Testcases for ambiguities
#[ 1000,0,1],
#[ 5, 0, 1000 ],  
[ 5, 0, 1 ],
[ 5, 0, 1 ],
[ 5, 0, 1 ],
[ 5, 0, 1 ],
[ 5, 0, 1 ],
[ 5, 0, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ 0, -5, 1 ],
[ -23, -64, 0 ],
[ -5, 0, 1 ],
[ -5, 1, 1 ],
[download]

(
  [-1, 5, 1],
  [0, 10, 1],
  [1, 5, 1],
  [3, 4, 1],
  [5, 1, 1],
  [30, 0, 1],
  [0, -45, 1],
  [-23, -64, 0],
  [-5, 0, 1],
  [-5, 1, 1],
)
[download]

update

fixed final push @out, $sum;