The word "unique" means having a property that distinguishes it from all other things. In mathematics, this is meaningless if not followed by "such that" and the precise property that distinguishes it from everything else. Every $x$ is the unique $y$ such that $y=x$, so being unique without qualification is void of meaning.
The one useful use of "unique" in mathematics is therefore in the context "the is a unique $x$ (in a given set $X$) such that the property $P(x)$ holds", which translated into a logical formula gives $\exists x\in X:\bigl(P(x)\land \forall y\in X:P(y)\to x=y\bigr)$.
The word "distinct" applies to a a family of variables (often just two or three): the variables $x_i$ for $i$ in some index set $I$ are all distinct whenever $x_i=x_j$ implies $i=j$ (the map $I\to X:i\mapsto x_i$ is injective). It is not attached to universal quantification (saying "for all distinct $x$" is void of meaning), though the variables $x_i$ might have been introduced by quantification (either universal or existential, both are quite reasonable).
So there is nothing that links "unique" with "distinct", except that both are non-properties: they are void of meaning when applied (without qualification) to a single variable. I think what may be confusing you is the horrible abuse of "unique" that is sometimes used in combinatorics (or probability): determine the number of unique configurations of some kind. I never know what "unique" is supposed to mean in this context. It might mean "distinct" in the sense that each configuration is counted only once, but that is rather silly since this is implicit in counting things correctly. Sometimes "number of distinct" such and such may be used to indicate not attaching a multiplicity which the reader might otherwise be inclined to do: the number of distinct roots of a polynomial, or eigenvalues of a matrix. However one would never say "number of unique" in this context. More likely the "number of unique configurations" means count only classes for some (not explicitly indicated) equivalence relation of configurations, such as orbits for a symmetry group acting on the set of configurations (seatings around a round table...).
In any case "for all unique $x,y,z$" is meaningless. One could say "for all distinct $x,y,z$", which is a slightly informal way of saying for all $x,y,z$ such that $x\neq y\neq z\neq x$ (don't forget the last inequality!).