There's an assertion on Wikipedia that a trace monoid is a syntactic monoid because $x w y \equiv x v y$ implies that $w \equiv v$. I don't see how that follows as a consequence, and I can't find any mention of this claim outside of Wikipedia and its copies.
Now, I can find claims and proofs that the free monoids are syntactic monoids. The relevant language is called a disjunctive set (or disjunctive language), and I have found a recipe for disjunctive sets for a free monoid $X^*$ (with $|X| \geq 2$) - any 'semi-discrete' 'dense' set will serve. $L$ is 'semi-discrete' if there is a constant bound on the number of its elements of each length, and is 'dense' if the bi-ideal $X^* u X^*$ meets $L$ for each string $u$. I haven't found an analogue to this recipe for traces, though it might work if one works with the language in the trace monoid rather than the free monoid - the disjunctive set in the free monoid cannot be semi-discrete for a non-trivial independence relationship. The proofs for the free monoid case lean heavily on 'primitive' strings (i.e. those that aren't multiple repeats of some other string), and the basic results for them fail heavily for traces. One might get somewhere with connected traces (those not writable as $vu=uv$), but I am not sure that such an approach would work.
I have made some slight progress - the direct product of disjunctive sets for different trace monoids is a disjunctive set for the direct product of the trace monoids, so the class of trace monoids that are syntactic monoids is closed under direct product.
I've made some progress in aping Proposition 2.2 of Reis and Shyr's Some properties of disjunctive languages on a free monoid. The stumbling block is to show that $\forall w \in X^*, xwy=ywx$ implies that $x$ and $y$ have the same length. This assertion isn't always true - it fails if $y = xz$ and $z$ commutes with everything. However, if there is a letter that commutes with nothing, and $|X| \geq 2$, then it does hold and so dense semi-discrete sets in the trace monoid are disjunctive. Thus I've now shown that the trace monoid of Unicode strings under canonical equivalence is a syntactic monoid.