Consider the differential equation $\dot{x} = \sin x$. The stability and dynamics of this equation has been discussed thoroughly in Nonlinear Dynamics and Chaos by Strogatz.
If I change the equation into a recursive relation $x_{t+1} = \sin(x_t)$, of course notions of stability are still present in this form even in discrete $t$.
My question is: is it possible that the techniques/analysis used in differential equations be used in recursive relations? Are there parallelism, similarities?
An appropriately placed Poincare section can take the system from a continuous time to a discrete time description. Now let us look at your question using a simple example which brings out the correspondence between these two descriptions. Considering a periodic solution of a differential equation (continuous time) and its counterpart in a discrete time system (given you can define an appropriate Poincare section) which is a stationary solution/fixed point.
To determine the stability of this solution in a continuous time formalism, you will need to calculate the Floquet exponents for this periodic solution. If all these exponents (excluding the vanishing longitudinal exponent) are strictly negative, then the periodic solution is stable, and unstable in case of positive exponents. Correspondingly for the discrete time system, say given by $x_{n+1}=f(x_n)$, one iteration stationary solutions (which correspond to a periodic solution of the differential equation from which the map was derived) are given by solving the equation $x_{n+1}=x_n \implies x^*=f(x^*)$ where $x^*$ is the stationary solution. The stability of $x^*$ is determined by the derivative on the RHS of the discrete time system at $x^*$, i.e. $f'(x^*)$ (where $f'=df/dx$), and $x^*$ is stable if $|f'(x^*)|<1$ and unstable if $|f'(x^*)|>1$.
Therefore, you can use either of these descriptions to determine the stability of periodic solutions. Although this answer is quite crude but I hope it brings out the parallelism more clearly for you.