The two empires did not fall simultaneously. The reason the
Abbassid Caliphate fell in the Middle East was because the Mongol
Empire defeated it and installed the Ikhanate. As a result, the
political state of the Middle East following the fall of the
Abbassids was the Ikhanate in most areas, the Mamluks in the Levant
and Egypt, and the untouched Arabian Emirates (like the Sharifate
of Mecca, the Emirates of the modern UAE, Oman, etc.)
The fall of the Ikhanate in 1341 led to the development of many
smaller piecemeal kingdoms throughout the former territories of the
Ikhanate. The smaller states in Persia were conquered relatively
quickly by the Timurids of Transoxiana. Elsewhere in the Middle
East, these smaller states would later be conquered by the Ottoman
Empire. The Mamluks held their position in the Levant until the
Ottomans eventually conquered them as well. The Arabian Emirates
remained untouched.