Bjorn Poonen addresses this question for commutative (associative, unital) algebras in The moduli space of commutative algebras of finite rank; asymptotically we have
$$q^{\frac{2}{27} n^3 + O(n^{8/3})}$$
such algebras (Theorem 10.9). 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 An estimate of the number of parameters defining an $e$-dimensional algebra 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.