We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields $X$ on ${\mathbb{R}}^2$ written as stochastic integral with respect to infinitely divisible random measure. The scaling procedure involves increments of $X$ over points the distance between which in the horizontal and vertical directions shrinks as $O(λ)$ and $O(λ^γ)$ respectively as $λ\downarrow 0$, for some $γ>0$. We consider two types of increments of $X$: usual increment and rectangular increment, leading to the respective concepts of $γ$-tangent and $γ$-rectangent random fields. We prove that for above $X$ both types of local scaling limits exist for any $γ>0$ and undergo a transition, being independent of $γ>γ_0$ and $γ<γ_0$, for some $γ_0>0$; moreover, the "unbalanced" scaling limits ($γ\neγ_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or $1$. The paper extends Pilipauskaitė and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on ${\mathbb{Z}}^2$ and Benassi et al. (2004) on $1$-tangent limits of isotropic fractional Lévy random fields.