one cannot generally assign an order to the Schrödinger-Equation
This statement is completely incorrect. If you mean a "Schrodinger equation that describes all systems" and so the order cannot be defined for all systems, your statement is still not true since even though there are other "renderings" of it that are for specific systems, you can't really call them the Schrodinger equation in such cases.
For simple non-relativistic systems, the standard time independent Schrodinger equation is second-order in space derivatives and the time dependent Schrodinger equation is the same but with also a first order time derivative. These equations are applicable to many quantum systems. For other systems:
If one considers the spin of an electron
When we consider (relativistic) spin $\frac 12$ particles, we can use the Dirac equation, which is of the form $$\left(i\hbar\gamma^\mu\partial_\mu-mc\right)\psi=0$$ We see that this equation is first order in space and time (there is of course the Pauli equation for non-relativistic electrons in magnetic fields which is second order in space and first order in time).
In relativistic but spinless systems, we can use the Klein-Gordon equation $$(\Box^2-m^2)\psi=0$$ (in units where $c=\hbar=1$), which also is second order in space derivatives and also second order in time derivatives.