Estimates based on key features of the film - including geographic landmarks, government structure, technological progress, linguistic terms, and religious references - place Disney's Aladdin as set in Central Asia (such as modern day Iraq or Uzbekistan) in the 900s AD. In this film, two major side-characters are the animal sidekicks of the protagonist and the villain- Abu and Iago. Their species are generally identified as a gray langur monkey and a scarlet macaw, however neither of these species are endemic to the proposed region. The gray langur genus is native to the Indian subcontinent while the scarlet macaw is much farther away, ranging from Mexico to Brazil. Would there have been trade routes, however distant, between these areas and the proposed region of the setting that could explain the presence of these two animals so far from their native lands?
Well, first of all a major geographic problem is that Central Asia which you mention does not refer to Iraq. Central Asia comprises the group of 5 countries (Kazakhstan, Kyrgyzstan, Tajikistan, Uzbekistan and Turkmenistan) including the region of East Turkestan. Iraq is a part of the Middle East and apart from the presence of Islam in both countries neither has any strong cultural links with one another.
Secondly, there are no geographic distinctions that relate it to any modern day, or historical, place. Agrabah is a composition of various fantasies concocted by Orientalist ideas into a fantasical place. It is very roughly, and I do mean very roughly, set in some form of Baghdad like city but that is it. Trying to place it in any realistic setting cannot be done.
On to the question, trade routes existed between the Abbasid Caliphate and certain geographical regions of India in particular with Northern India and with Arab settlements which had existed in the Deccan and South India regions. There are no accounts of live animal trade that I know of but there were animal products that were going towards the Abbasid court. It is realistic to assume that Monkeys, especially Langurs, could be traded along these routes.
There were no trade routes open at this time to either Mexico or Brazil. It must be remembered that these are movies meant to entertain not to be historically accurate. The original story of Aladdin did not occur in a Middle Eastern setting, it was a mixture of a mythical China/East Turkestan and did not feature any animal companions.
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