Why in FSK the phase shift is allowed?
My first question is, why should we allow the signals have different phases?


Counterquestion: Why should they not be allowed? Enforcing continuous phase can make systems more complex, without any benefit.

I cant see a reason for this and as far as I know, different phases result in phase discontinuity of the transmit signal.

Depending on how you do that transition, that's not necessarily the case: you can pulse-shape $\theta_i$, or you can filter $x$.
It's less often a problem than old textbook make it out to be, to be honest.

(2) And the second question is, how this difference in phase is related to coherent and non-coherent receiver?

Not at all. What the receiver does says nothing about what the receiver does!