Here is some vague thought about 3d mirror symmetry (following Ben G and Justin H).

Let $G$ and $G^L$ be Langlands dual varieties. Assume $(G, M)$ is S-dual to $(G^L, M^L)$, where $M$ is some nice G-Ham space, same for $M^L$. Example $$ M = pt, \quad M^L= Whit_{G^L}(T^*G^L) $$ where $Whit_{G^L}$ is one-side symp reduction by $U$ with generic character.

We can consider endomorphism type spaces $$ M/G \times_{T^*[1]BG} M/G \leftrightarrow M^L/G^L \times_{T^*[1]BG^L} M^L/G^L$$ In this example, we should get what? The space $T^*(BG)$ 3d mirror to $Whit_{G^L}^2(T^*G^L)$.

OK, if $M$ is a usual 0-shifted symp stack, then $[M/G]$ should be viewed as a 1-shifted Lagrangian in the 1-shifted cotangent bundle $T^*[1]BG = [g^*/G]$, that's what G-Ham space gives you. So that the intersection is like 0-shifted symp space.

And the terminology about Coulomb branch, and bi-Whittaker reduction is that $$ M_C(G, M) = Whit_{G^L}(M^L) = Whit_{G^L}(T^*G^L) \times_{T^*[1]BG^L} M^L/G^L. $$ $$ M_H(G, M) = M / / G = 0/G \times_{T^*[1]BG} M/G $$ $$ M_H(G, M) \leftrightarrow M_C(G, M) $$

So Coulomb branch 3d mirror to Higgs branch is a special case of relative Langlands, where one slot is about the basic dual setup.