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).
Added as an edit after Alexandre's comment since I am not entitled to comment.
One way to construct the universal space is to take the free vector space over $U$ and provide it with the finest locally convex topology such that the embedding of $U$ is holomorphic, then take the completion.
I doubt that you will find the fact that the dual of $H(U)$ has the universal property in the literature (such considerations were never fashionable---too much category theory for the analysts, too much hard analysis for the category theorists perhaps). It follows very easily from the theory of duality for $H(U)$ (Köthe, Crelle (191)). An accessible version in english is in the book "Complex Analysis: a functional analysis approach" by Ruecking and Rubel. The vector-valued case is in the seminal follow-up papers to Köthe's by Grothendieck in Crelle, 192.
I should note that the duality mentioned above was originally developed by the portuguese mathematician J. Sebastião e Silva in a sadly forgotten article in Port. Math. 9 (1950) 1-130 and this again has its source in work by Cacciopoli and Fantappié. The universal property mentioned above has many analogues---e.g., the distributions on the unit interval, universal for smooth mappings into Banach spaces (with obvious generalisations), Radon measure on the unit interval or a compact space, universal for continuous mappings, bounded Radon measures on a completely regular space (bounded, continuous mappings), uniform measures on a uniform space (bounded, uniformly continuous mappings). See, for example Raikov, Math. Sb. 63 (1964) 582-590, Tomašek, Czech. Math. J. 20 (1970) 1-18, 19-33.