Why is an isothermal process not unique in nature?
(1) An isothermal process is one during which the system temperature $T=constant$. For an ideal satisfying $pV=RT$ this means that $pV=constant$ irrespective of how you achieve it. You can go up and down, back and forth on the same isothermal, but it is still an isothermal process having the same temperature. So it is not a unique process between two fixed points.

(2) The uniqueness of the adiabatic process is true only for a simple system, that is one whose state can be described by temperature and one additional "mechanical" parameter, e.g., for a gas $p = p(T,V)$, or a magnet $M=M(T,B)$, etc. It is definitely not true for more complicated systems, for example a gas with magnetic properties $p=p(T,V,B)$. Instead what is true is that through every equilibrium point there is a unique adiabatic surface (not a curve) determined by a pair of variables say $T,V$ or $V,B$ or $T,B$ such that any curve lying in that surface is an adiabatic process. That such adiabatic surfaces exist is the basis of one of the mathematical proofs of the existence of the entropy function and this quite famous proof is due to Caratheodory.