The celebrated result of Gortler-Healy-Thurston (independently, Jackson-Jordán for $d=2$) shows that the global rigidity of graphs realised in the $d$-dimensional Euclidean space is a generic property. Extending this result to the global rigidity problem in $\ell_p$-spaces remains an open problem. In this paper we affirmatively solve this problem when $d=2$ and $p$ is an even positive integer. A key tool in our proof is a sufficient condition for the $d$-tangential weak non-defectivity of projective varieties due to Bocci, Chiantini, Ottaviani, and Vannieuwenhoven. By specialising the condition to the $p$-Cayley-Menger variety, which is the $\ell_p$-analogue of the Cayley-Menger variety for Euclidean distance, we provide an $\ell_p$-extension of the generic global rigidity theory of Connelly. As a by-product of our proof, we also offer a purely graph-theoretical characterisation of the $2$-identifiability of an orthogonal projection of the $p$-Cayley-Menger variety along a coordinate axis of the ambient affine space.