Why in FSK the phase shift is allowed? FSK modulation encodes information into the frequency of the signals. But when I checked the reference books, the formal expression for a frequency modulated signal is:
$x_i(t) = \alpha cos(2\pi f_it + \theta_i)$
Which means, for a binary FSK, we can consider $x_1(t)$ and $x_2(t)$ with different frequencies $f_1$ and $f_2$ (which make sense) and different phase $\theta_1$ and $\theta_2$ (which is odd to me), respectively.
(1) My first question is, why should we allow the signals have different phases? I cant see a reason for this and as far as I know, different phases result in phase discontinuity of the transmit signal.
(2) And the second question is, how this difference in phase is related to coherent and non-coherent receiver?
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!