You have the correct implementation, but you're not checking both the eigenvector and eigenvalue for convergence. You're only checking the eigenvalue for convergence. The power method estimates both the prominent eigenvector and eigenvalue, so it's probably a good idea to check to see if both converged. When I did that, I managed to get [1/3 1 1/3]. Here is how I modified your code to facilitate this:

function [m,y_final]=power_method(A,x)
m=0;
n=length(x);
y_final=x;
tol=1e-10; %// Change - make tolerance more small to ensure convergence
while(1)
     mold = m;
     y_old=y_final; %// Change - Save old eigenvector
     y_final=A*y_final;
     m=max(y_final);
     y_final=y_final/m;
     if abs(m-mold) < tol && norm(y_final-y_old,2) < tol %// Change - Check for both
         break;
     end
end
end

When I run the above code with your example input, I get:

>> [m,y_final]=power_method(A,x)

m =

     2


y_final =

    0.3333
    1.0000
    0.3333

On a side note with regards to eig, MATLAB most likely scaled that eigenvector using another norm. Remember that eigenvectors are not unique and are accurate up to scale. If you want to be sure, simply take the first column of V, which coincides with the dominant eigenvector, and divide by the largest value so that we can get one component to be normalized with the value of 1, just like the Power Method:

>> [V,D] = eig(A);
>> V(:,1) / max(abs(V(:,1)))

ans =

    0.3333
    1.0000
    0.3333

This agrees with what you have observed.