Bjorn Poonen addresses this question for commutative (associative, unital) algebras in *<a href="https://arxiv.org/abs/math/0608491">The moduli space of commutative algebras of finite rank</a>*; asymptotically we have

$$q^{\frac{2}{27} n^3 + O(n^{8/3})}$$

such algebras (Theorem 10.9). Bjorn also gives a more precise lower bound on the dimension of the corresponding affine scheme in Theorem 9.2 which is a collection of three polynomials with leading term $\frac{2}{27} n^3$ depending on the value of $n \bmod 3$. The $\frac{2}{27}$ may seem familiar from a corresponding count of the number of finite $p$-groups and it happens for very similar reasons as he discusses in Section 10:

> The approach towards both those results is to adapt the proof (begun in [Hig60] and completed in [Sim65]) that the number of $p$-groups of order $p^n$ is $p^{ \frac{2}{27} n^3 + O(n^{8/3})}$. As suggested to us by Hendrik Lenstra, there is an analogy between the powers of the maximal ideal of a local finite-rank $k$-algebra and the descending $p$-central series of a $p$-group. Although there seems to be no direct connection between finite-rank $k$-algebras and finite $p$-groups, the combinatorial structure in the two enumeration proofs are nearly identical.

He also cites *<a href="http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1295&option_lang=eng">An estimate of the number of parameters defining an $e$-dimensional algebra</a>* by Yuri Neretin (which is in Russian, sadly for me) as addressing the Lie and associative cases; I'm not sure if the estimates immediately carry through to a finite field but if they do the answer is the same for Lie algebras and for associative algebras it is

$$q^{ \frac{4}{27} n^3 + O(n^{8/3}) }.$$

Presumably the analogous structure for Lie algebras making the answer similar is the descending central series for a nilpotent Lie algebra. For the associative case maybe it's something like powers of the Jacobson radical? 

Note also that because $\frac{8}{3} > 2$ the error term in the exponent absorbs multiplicative factors as large as $q^{O(n^2)}$ so these asymptotics hold regardless of whether or not we quotient by the action of $GL_n(\mathbb{F}_q)$ (which is equivalent to asking for the isomorphism classification), which you might see as unsatisfactorily lenient but I think these are state of the art.