Octave, 453

format plus 'XO.'
disp(F=zeros(6))
r=@()fread(0,1);R=@()r()-47;q=@(m)printf(m)&&quit;e=@()q("INVALID");l=@(n)n<1||n>6&&e();s=@()r()==32||e();w="P1 WINS";f=@(v)abs(sum(v))==22.5&&q(w);g=@(m)any(abs(sum(m))==22.5)&&q(w);d=@diag;while 1
i=R();i==66&&quit;l(i);s();l(j=R());s();F(j,i)&&e();abs(v=R()-36.5)==4.5||e();F(j,i)=v;disp(F)
r();f(d(F,-1));f(d(F,0)(2:6));f(d(F,0)(1:5));f(d(F,1));g(F(1:5,:));g(F(2:6,:));g(F(:,1:5)');g(F(:,2:6)');F&&q("P2 WINS");end

The implementation is quite straightforward, the only real “trick” to use format plus to take care of printing. Here is a commented version:

format plus 'XO.'               # this is where the magic happens

## initialize and print empty board
disp(F=zeros(6))

## shortcuts
r=@()fread(0,1);
R=@()r()-47;
q=@(m)printf(m)&&quit;
e=@()q("INVALID");
l=@(n)n<1||n>6&&e();
s=@()r()==32||e();
w="P1 WINS";
f=@(v)abs(sum(v))==22.5&&q(w);
g=@(m)any(abs(sum(m))==22.5)&&q(w);
d=@diag;

while 1
  i=R();                        # read index 1
  i==66&&quit;                  # ‘q’?
  l(i);                         # check bounds
  s();                          # read and check space
  l(j=R());                     # read and check index 2
  s();                          # read and check space
  F(j,i)&&e();                  # square already filled?
  abs(v=R()-36.5)==4.5||e();    # valid mark?
  F(j,i)=v;                     # assign …
  disp(F)                       # and print board
  r();                          # read off newline

  ## check diagonals
  f(d(F,-1));
  f(d(F,0)(2:6));
  f(d(F,0)(1:5));
  f(d(F,1));

  ## check rows
  g(F(1:5,:));
  g(F(2:6,:));

  ## check columns
  g(F(:,1:5)');
  g(F(:,2:6)');

  ## check chaos
  F&&q("P2 WINS");
end

I took care to output the board and messages in a tidy manner. If some extra output is acceptable, I could shave off a couple of bytes. For example, using automatic printing (no disp(F)), the board would be shown as

F =

......
......
......
......
......
......

Also, I interpreted that each move is given on a separate line.