It depends on what counts as "discovery". Interestingly, optical anisotropy was discovered before the elastic one, and the first anisotropic "material" modeled was... the luminiferous ether.
Bartholinus discovered double refraction in calc-spar, a type of calcite, back in 1669, and Huygens showed in 1690 that two rays arising from refraction by calcite are extinguished by passing them through the same crystal rotated about the direction of the rays. Thus, calcite was optically anisotropic. Newton explained it by suggesting that light particles have "sides", and used it as an argument against Hooke's wave optics. At the time, only longitudinal (acoustic) waves were known, and they could not explain transverse polarization.
The phenomenon was not much investigated until Malus's polarization experiments with calcite in 1808, and Arago's with quartz in 1811, see Brief History of the Discovery of Phenomena Concerning Light Polarization. At that time, Young and Fresnel made wave optics indispensable, and that posed a problem. In isotropic elastic media transverse waves were always accompanied by longitudinal waves, but polarization experiments ruled out their presence. If light propagated in elastic ether, that ether had to be anisotropic. Elastic theories of ether became quite popular, with Cauchy, Lame, Green and others studying them. Green's 1838 equations already took anisotropy into account, but it was Kelvin's Elements of a Mathematical
Theory of Elasticity (1856) that redirected the theory towards real solid materials rather than the hypothetical ether, see 75-plus years of anisotropy by Helbig and Thomsen:
"Thus, the first articles on elasticwave
propagation already took anisotropy into account. For
example, Green (1838) was the first to use strain energy, and
he strongly supported the notion that there could be as many
as 21 elastic constants. In 1856, Lord Kelvin published “Elements of a mathematical theory of elasticity”, which exclusively discussed solids.
This is not to be taken as an indication that he did not believe
in the elastic ether, but only that he was interested in metals at
that time and thus needed a solid foundation of the theory of
For this purpose, he invented concepts that became
common only much later, such as vectors and vector spaces
(in 6D space!), tensors, and eigensystems. With these tools, he
could describe the elastic tensor in coordinate-free form. His
ideas were so much ahead of his time that his paper — and a
re-publication (in the “Elasticity” listing of the 1886 edition of
the Encyclopedia Britannica) — were regarded by some of his
contemporaries as scientifically unsound (despite his stature)
and thus made no impact on the development of the theory of
anisotropy... Kelvin was also the first to formulate the elastic-wave equation for anisotropic media. (He solved it for a simple case.)
Since this achievement was published as part of his “no impact”
papers, it was also overlooked. Hence, today the solution
of the wave equation is attributed to Christoffel (1877)."
In the end, anisotropic ether did not work out either, but the elasticity theory motivated by it came in handy in material science.