There are some plausible arguments that the authors of the Iliad and Odyssey are not the same, even if they must have been closely related.

For instance I find very plausible the hypothesis that the Iliad was composed by a man and the Odyssey was composed by a woman.

The main reason is the striking resemblance of the Iliad with an action movie, in contrast with the Odyssey that looks exactly like a chick flick.

More seriously, the Iliad focuses on a lot of things of great interest for the men of those times, like gory details about the best methods to kill or maim opponents in combat, or athletic competitions, while the Odyssey focuses on many things of interest for females, like clothes, food, gardens, romance, and it even has several passages that look quite feminist, despite being written millennia ago, by complaining about the discrimination unjustly enforced by men against women. The Odyssey also has a lot of strong female characters that are more important than the men, e.g. Calypso, Circe, Aurora, Nausicaa, Penelope.

The text of the Odyssey also contains evidence of being composed later than the Iliad, perhaps by several decades (due to some evolution of the language). So it has been hypothesized that the Odyssey was composed by a female relative of Homer, e.g. by his daughter or niece.

In any case, the author of the Odyssey mastered perfectly the same kind of language and poetic structures that were used in the Iliad, so he or she must have studied for many years the older poems in the same tradition, like the Iliad and the many others that have been lost.

A very large number of verses and short sequences of verses from the Iliad and the Odyssey are much older than the complete poems. However, someone, i.e. "Homer" alone, or with his assistants or relatives, had integrated all the older inherited poetic fragments into big coherent epic poems, before they were recorded in writing.