Well, one trivial connection is that if you look at $1\times 1$ matrices (which have only a single complex entry), then you'll find that it is real iff it is Hermitian, its complex conjugate is its conjugate transpose, and its polar decomposition is the polar form.
Also, just like a complex number can be uniquely decomposed into a real and an imaginary part ($z = a+\mathrm ib$ with real $a,b$), a complex matrix can be uniquely decomposed into a Hermitian and an "anti-Hermitian" part, i.e,. $M =A + \mathrm iB$ with $A$ and $B$ Hermitian. And just like $\Re(z)=\frac12(z+\bar z)$ and $\Im(z)=\frac1{2\mathrm i}(z-\bar z)$, the Hermitian part of a matrix is $\frac12(M+M^*)$ and the "anti-Hermitian" part is $\frac1{2\mathrm i}(M-M^*)$.
Moreover, just like $\bar zz$ is a non-negative real number, $M^*M$ is a positive semidefinite matrix.
Another point: Hermitian matrices have real eigenvalues, and unitary matrices have eigenvalues of the form $\mathrm e^{\mathrm i\phi}$.
About the usefulness of the analogy:
In classical physics, observables should be real. In quantum physics, observables are represented by Hermitian matrices. Also, the quantum analogue to probability densities, which are non-negative functions with integral $1$, are density operators, which are positive semidefinite matrices with trace $1$. So there's indeed some connection.
Since $A^T = \overline{A}$ and using basic properties of trace :
$$ tr(AB) = tr((AB)^T) = tr(B^T A^T) = tr(\overline{B} \ \overline{A})= tr(\overline{A} \ \overline{B}) $$
Therefore :
$$ tr(AB) = tr(\overline{A} \ \overline{B}) = tr(\overline{AB}) = \overline{tr(AB)} $$
Finally $tr(AB) \in \mathbb R$.
Best Answer
By assumption, there exist two matrices $C,D$ sucht that $$ A=C^*C\quad\mbox{and}\quad B=DD^*. $$ Now, using the commutativity of the trace, $$ \mbox{Tr}(AB)=\mbox{Tr}(C^*CDD^*)=\mbox{Tr}(D^*C^*CD)=\mbox{Tr}((CD)^*CD) $$ appears to be the sum of of all $|(CD)_{i,j}|^2$.
So $$ \mbox{Tr}(AB)=0\quad\Rightarrow \quad CD=0\quad \Rightarrow\quad AB=C^*CDD^*=0. $$
Note: this proof works in the complex case, like in the real case.