  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 $O(\lambda)$ and
$O(\lambda^\gamma)$ respectively as $\lambda \downarrow 0$, for some
$\gamma>0$. We consider two types of increments of $X$: usual increment and
rectangular increment, leading to the respective concepts of $\gamma$-tangent
and $\gamma$-rectangent random fields. We prove that for above $X$ both types
of local scaling limits exist for any $\gamma>0$ and undergo a transition,
being independent of $\gamma>\gamma_0$ and $\gamma<\gamma_0$, for some
$\gamma_0>0$; moreover, the "unbalanced" scaling limits ($\gamma\ne\gamma_0$)
are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or
$1$. The paper extends Pilipauskait\.e 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 L\'evy
random fields.
