Edit: Following discussion with Tomasz below, I want to clarify that I mean for definable subsets here to range over arbitrary sorts; ie, $\mathfrak{C}$ can be multi-sorted, but I'm not considering it as having a distinguished "home sort" or using the term "definable subsets" to refer to subsets of a distinguished sort.
Otherwise, as he points out, one can consider the theory of infinite sets as a multi-sorted theory $T^*$, with sorts $(S_n)_{n\in\omega}$ and $n$-ary projection maps $\pi_n:S_1^n\to S_n$, where $S_1$ is taken to be the "home sort" and each $S_n$ is taken to be the sort for the subsets of $S_1$ of size at most $n$. Then any finite subset $\{x_1,\dots,x_n\}\subset S_1$ of the home sort has a canonoical parameter given by $\pi_{n}(x_1,\dots,x_n)$, but (as Tomasz argues) the pair of tuples $\{(a,b),(c,d)\}\subset S_1^2$ for distinct elements $a,b,c,d\in S_1$ has no canonical parameter.
However, this is not quite what I'm looking for, since $T^*$ only has elimination of $1$-imaginaries for subsets of the home sort. Indeed, for instance, the finite subset $\{\pi_2(a,b),\pi_2(c,d)\}\subset S_2$ has no canonical parameter, so $T^*$ does not eliminate $1$-imaginaries for subsets of arbitrary sorts.
Let $T$ be a complete first-order theory and fix a monster model $\mathfrak{C}\models T$. Recall that, for a $\mathfrak{C}$-definable collection of $n$-tuples $E\subseteq\mathfrak{C}^n$, a canonical parameter for $E$, denoted $\ulcorner E\urcorner$, is a finite tuple $\overline{e}\in\mathfrak{C}$ such that an automorphism of $\mathfrak{C}$ fixes $E$ setwise if and only if it fixes $\overline{e}$; this is unique up to interdefinability. Now, $T$ eliminates imaginaries if every $\mathfrak{C}$-definable set of $n$-tuples has a canonical parameter, and $T$ eliminates finite imaginaries if every finite set of $n$-tuples has a canonical parameter.
Example: If $T$ is the theory of infinite sets, then $T$ does not eliminate finite imaginaries, since for any $c_1\neq c_2\in\mathfrak{C}$ the set $\{c_1,c_2\}$ has no canonical parameter.
Example: If $T$ is totally ordered, then $T$ eliminates finite imaginaries, since an automorphism fixes a finite collection of $n$-tuples setwise if and only if it fixes it pointwise.
Now, I'm curious about how much elimination of imaginaries can fail for tuples as opposed to elements. Say that $T$ eliminates $1$-imaginaries if every $\mathfrak{C}$-definable subset of $\mathfrak{C}$ has a canonical parameter, and say that $T$ eliminates finite $1$-imaginaries if every finite subset of $\mathfrak{C}$ has a canonical parameter.
Question: Is there a theory that eliminates $1$-imaginaries but does not eliminate imaginaries? Is there a theory that eliminates finite $1$-imaginaries but does not eliminate finite imaginaries?
I suspect the answer is yes, but I'm struggling to come up with any examples.
