Kinds of uniqueness
Definitions
Say we define a value $x$ by giving a condition $\phi$ that must hold on it. Then we say:
$x$ is unique if the condition $\phi$ determines at most one value. Mathematically, we consider $x$ to be unique if for all $a, b$ satisfying $\phi$ we have that $a = b$
$x$ is unique up to isomorphism if all values satisfying $\phi$ are isomorphic to each other Likewise, this is expressed as $\phi(a) \land \phi(b) \implies a \cong b$
Generally for any equivalence relation $\sim$ we say that $x$ is unique up to $\sim$ if all values satisfying $\phi$ are related by $\sim$ Expressed as $\phi(a) \land \phi(b) \implies a \sim b$
Examples
Define $x$ to be the value $x \in \mathbb R$ such that $x^2 = 2$. Then $x$ is not unique, since both $\sqrt 2$ and $-\sqrt 2$ satisfy this condition.
Define $x$ to be the value $x \in \mathbb R$ such that $x > 0$ and $x^2 = 2$. Then $x$ is unique and is given by $x = \sqrt 2$
Define $x$ to be the 0-element set. Then $x$ is unique and is given by $x = \varnothing$.
Define $x$ to be the 1-element set. Then $x$ is not unique but is unique up to isomorphism, since all 1-element sets are isomorphic (wrt the category $\textbf{Set}$)

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