Here is a method of recovering the topology of $H(U)$ from general considerations.
The idea is that the dual of $E$ of $H(U)$ has the following universal property: $E$ is a complete locally convex space (even a so-called nuclear Silva space, i.e., an inductive limit of a sequence of Banach spaces with nuclear intertwining mappings) and $U$ embeds in $E$ in such a manner that every holomorphic mapping from $U$ into a Banach space lifts in a unique manner to a continuous linear mapping on $E$. We now forget the topology on $H(U)$ and note that the existence of such a universal space can be proved without recourse to this duality (this is a standard construction as a closed subspace of a suitable large product of Banach spaces---analogous to the construction of the free locally convex space over a completely regular space or a uniform space---see, e.g. Raikov, Katetov, etc.) Such a free object is always unique in a suitable sense. Now it follows from the universal property (applied to scalar-valued functions) that $H(U)$ is, as a vector space, naturally identifiable with the dual of the universal space. It can then be provided with the corresponding strong topology which is thus intrinsic. But this is precisely the standard Fréchet space topology (the fact that we are dealing with a symmetric duality between a nuclear Fréchet space, resp. Silva space is relevant here).