Does every closed subset in a scheme correspond to a unique closed subscheme? Could anyone give an example of a closed subset $Y$ in a scheme $X$,
such that there is no closed immersion from a scheme into $X$ whose "image of underlying points" is $Y$?
Also, could anyone give an example $Y$ that admits more than one "realization" as a closed subscheme of $X$?
 A: Every closed subset can be made into a closed subscheme, e.g. by choosing the reduced structure (see for example tag 01J3 in the stacks project). This would be the canonical choice of a scheme structure on the given closed subset, but there are many more in general (see below for an example).
Regarding your second question:
Just take for example $X = \mathbb{A}^1_x$ and the closed subschemes $Y = V(x) = \text{Spec}(k[x]/(x))$ and $Y' = V(x^2) = \text{Spec}(k[x]/(x^2))$. In this case $Y$ is the reduction of $Y'$. Higher powers of $x$ all define closed subschemes of $\mathbb{A}^1$ with the same underlying topological space (which is just the origin). Here you can already see that there are plenty such subschemes.
In classical algebraic geometry one is (depending on the settup...) not necessarily able to distinguish these subschemes (subvarieties), but they occur very naturally (you would certainly want to know multiplicities of roots of a given polynomial instead of just knowing the set of roots). This is one of the reasons for introducing structure sheaves. They are in this regard so to say the glasses we need to see properly.
