A regular 1-category is exact when every congruence has a kernel. The definition for 2-categories is analogous.
Let $-1<n\le 2$ be directed (see n-prefix). A 2-category is $n$-exact if it is regular and every n-congruence is a kernel.
Recall that a 1-category is regular as a 1-category iff it is regular as a homwise-discrete 2-category. However, a regular 1-category is exact as a 1-category precisely when it is 1-exact as a 2-category. Since general 2-congruences in a 1-category are internal categories, they will obviously not all be kernels. This is why we add a prefix to “exact:” the best notion for 2-categories, which we call “2-exact,” is not a conservative extension of the established meaning of “exact” for 1-categories.
Likewise, it is unreasonable to expect a (2,1)-category to be any more than (2,1)-exact, or a (1,2)-category to be any more than (1,2)-exact.
Cat is 2-exact. Likewise, Gpd is (2,1)-exact, Pos is (1,2)-exact, and of course Set is 1-exact.
Every regular (0,1)-category (that is, every meet-semilattice) is (0,1)-exact, and in fact even 2-exact, since there are no nontrivial congruences of any sort in a poset.
If $K$ is 2-exact, then by the classification of congruences, $\mathrm{gpd}(K)$ is (2,1)-exact, $\mathrm{pos}(K)$ is (1,2)-exact, $\mathrm{disc}(K)$ is 1-exact, and $\mathrm{Sub}(1)$ is (0,1)-exact.
If $K$ is $n$-exact, then so is ${K}^{\mathrm{co}}$, by the remarks about opposite 2-congruences. For 1-categories and (2,1)-categories, of course, this is contentless, but for 2-categories and (1,2)-categories it is contentful.