You should have a look at Chapter 8 of Lange and Birkenake *Complex Abelian Varieties*. I will refer to this book in what follows.

Fix a type $D= \textrm{diag}(d_1, \ldots, d_g)$. Then the moduli space of polarized complex abelian varieties of dimension $g$ and type $D$ with symplectic basis is precisely the Siegel upper half-plane $\mathfrak{H}_g$. So this moduli space admits a universal family, that is constructed in Section 8.7.

If you want to forget the symplectic basis, then you need to consider the moduli space $\mathcal{A}_D=\mathfrak{H}_g/G_D$, where $G_D$ is a suitable discrete subgroup of $\textrm{Sp}_{2g}(\mathbb{R})$. The space $\mathcal{A}_{D}$ is a normal, complex analytic space of dimension $\frac{g(g+1)}{2}$, which is called the moduli space of polarized abelian varieties of type $D$. Since the quotient of an algebraic variety under a good action is also an algebraic variety, it follows that $\mathcal{A}_{D}$ is a complex algebraic variety. However, it is not a fine moduli space, because of the existence of polarized abelian varieties with extra automorphisms, so one in general does not expect a universal family over $\mathcal{A}_g$.

In order to obtain a fine moduli space it is necessary to rigidify our objects. In the analytic category one can choose to consider symplectic bases, in the algebraic category the rigidification can be made by using the so-called *level structures*.

You can find (much) more details in the book by Birkenhake-Lange.