1. yes, you need to use the == (double equal sign) instead of assignment
2. make sure you use the Clear at the beginning of your program so you don't "memorize" your mistakes
3. you call NDSolve twice but you should ca;ll it only once and solve for x and y simultaneously

You need to have both equations in a single DSolve function--otherwise there is no way for DSolve to know that there are two simultaneous equations. Also note that your wx'[t] should be w x'[t]: i.e., a space between the w and the x. Similarly for the wy'[t].

With that said, here is the form for DSolve that you want to use for these two simultaneous differential equations in x[t] and y[t]:

DSolve[
{y''[t] == w^2 y[t] - 2 w x'[t], 
  x''[t] == w^2 x[t] + 2 w y'[t]}, 
{y[t], x[t]}, t]

This gives a result in terms of unknown integration constants C[1], C[2], C[3], and C[4]. If you have initial conditions you can include them in the first argument list of the DSolve and the integration constants will be determined in the soluttion.