I can't speak to Jewish interpretation, but ancient Christians would not have understood "genre" in the sense that we do today. To them, all scripture was allegorical and all scripture was historical.
Jerome, in his Commentary on Jonah first reminded his readers that Jesus referred to Jonah typologically, and that this symbolism is the primary meaning of Jonah for Christians:
And Jonah was in the belly of the fish three days and three nights. LXX: 'and Jonah was in the belly of the whale for three days and three nights'. The Lord shows in the Gospel the symbolism of this passage, and it is superfluous to say in the same terms or even in other terms what he who has suffered has already said.
But Jerome argues that this story should be understood not just as a type but as a historical event, with no more reason to reject it than other miracle stories that were commonly accepted in Roman society at the time:
Then Jonah prayed unto the LORD his God out of the fish's belly. LXX: similar. If Jonah is compared to the Lord, and his time of three days and three nights in the belly of the whale is a sign of the suffering of the Saviour, his prayer also ought to be a kind of prayer of the Saviour. Some people, I don't doubt, will find it difficult to believe that a man can spend three days and three nights in the belly of a whale, especially after a shipwreck. These people can either be religious or not. But if they have faith, they will believe this all the more: how three children thrown into a furnace of hot fire were so well protected that their clothes were not even singed; how the sea drew back on itself into two sides and held itself up like a wall to offer a route for the people who wanted to pass; how with all human moderation the anger of a lion that had been increased by hunger was taken by fear at the sight of his prey, and didn't want to touch it; and even other such miracles. If they do not have faith, let them read the fifteen books of Ovid's Metamorphoses, and all Greek and Latin history. Therein they will see Daphne changed into a bay-tree, or the sister of Phaeton changed into poplars; how Jupiter the highest god, was transformed into a swan, flowed in gold and became a raging bull, and other adventures where the ugliness of the stories attest the holiness of the divinity. They believe in these stories and say that everything is possible for one god. And while they believe these ugly stories and defend the absolute power of a god, they do not attribute this same power to honest deeds.
Augustine, in a letter to Deogratias, agreed with Jerome that the book of Jonah described a miraculous event:
The last question proposed is concerning Jonah, and it is put as if it were not from Porphyry, but as being a standing subject of ridicule among the Pagans; for his words are: "In the next place, what are we to believe concerning Jonah, who is said to have been three days in a whale's belly? The thing is utterly improbable and incredible, that a man swallowed with his clothes on should have existed in the inside of a fish. If, however, the story is figurative, be pleased to explain it. Again, what is meant by the story that a gourd sprang up above the head of Jonah after he was vomited by the fish? What was the cause of this gourd's growth?" Questions such as these I have seen discussed by Pagans amidst loud laughter, and with great scorn.
To this I reply, that either all the miracles wrought by divine power may be treated as incredible, or there is no reason why the story of this miracle should not be believed. The resurrection of Christ Himself upon the third day would not be believed by us, if the Christian faith was afraid to encounter Pagan ridicule. Since, however, our friend did not on this ground ask whether it is to be believed that Lazarus was raised on the fourth day, or that Christ rose on the third day, I am much surprised that he reckoned what was done with Jonah to be incredible; unless, perchance, he thinks it easier for a dead man to be raised in life from his sepulchre, than for a living man to be kept in life in the spacious belly of a sea monster.
But Augustine agreed that the story was primarily typological:
It is neither unreasonable nor unprofitable to inquire what these miracles signify, so that, after their significance has been explained, men may believe not only that they really occurred, but also that they have been recorded, because of their possessing symbolical meaning.…
As to the question, What was prefigured by the sea monster restoring alive on the third day the prophet whom it swallowed? why is this asked of us, when Christ Himself has given the answer, saying, "An evil and adulterous generation seeketh after a sign, and there shall no sign be given it but the sign of the prophet Jonas: for as Jonas was three days and three nights in the whale's belly, so must the Son of man be three days and three nights in the heart of the earth"?
Augustine also suggested that the historicity of Jonah was not as important as certain other matters:
I have answered to the best of my power the questions proposed; but let him who proposed them become now a Christian at once, lest, if he delay until he has finished the discussion of all difficulties connected with the sacred books, he come to the end of this life before he pass from death to life. For it is reasonable that he inquire as to the resurrection of the dead before he is admitted to the Christian sacraments. Perhaps he ought also to be allowed to insist on preliminary discussion of the question proposed concerning Christ—why He came so late in the world's history, and of a few great questions besides, to which all others are subordinate. But to think of finishing all such questions as those concerning the words, "In what measure ye mete, it shall be measured unto you," and concerning Jonah, before he becomes a Christian, is to betray great unmindfulness of man's limited capacities, and of the shortness of the life which remains to him. For there are innumerable questions the solution of which is not to be demanded before we believe, lest life be finished by us in unbelief.
So in summary:
- Ancient people understood as well as we do that nobody could survive three days in the belly of a fish.
- Christians, however, maintained that with God anything was possible, and therefore Jonah could have survived being swallowed by the fish.
- But they also understood Jonah allegorically because Jesus did, as a foreshadowing of his own death and resurrection.
- And ultimately, Jonah's historicity shouldn't be treated as a stumbling block; if that's the only thing standing between the seeker and faith, let them believe first and unravel the mystery later.