  Let $B_n$ and $P_n$ be the unit ball and the unit polydisk in $\mathbb{C}^n$
with $n\geq 2$ respectively. Denote $\mbox{Aut}(B_n)$ and $\mbox{Aut}(P_n)$ the
holomorphic automorphism group of $B_n$ and $P_n$ respectively. In this paper,
we prove that $B_n$ admits no $\mbox{Aut}(B_n)$-invariant strongly pseudoconvex
complex Finsler metric other than a constant multiple of the
Poincar$\acute{\mbox{e}}$-Bergman metric, while $P_n$ admits infinite many
$\mbox{Aut}(P_n)$-invariant complete strongly convex complex Finsler metrics
other than the Bergman metric. The $\mbox{Aut}(P_n)$-invariant complex Finsler
metrics are explicitly constructed which depend on a real parameter $t\in
[0,+\infty)$ and integer $k\geq 2$. These metrics are proved to be strongly
convex K\"ahler-Berwald metrics, and they posses very similar properties as
that of the Bergman metric on $P_n$. As applications, the existence of
$\mbox{Aut}(M)$-invariant strongly convex complex Finsler metrics is also
investigated on some Siegel domains of the first and the second kind which are
biholomorphic equivalently to the unit polydisc in $\mathbb{C}^n$. We also give
a characterization of strongly convex K\"ahler-Berwald spaces and give a de
Rahm type decomposition theorem for strongly convex K\"ahler-Berwald spaces.
