The Mughals did behave as though they were Caliphs and asserted it explicitly, while not disputing that the Ottomans were too . . . the Ottomans didn't see it that way, but neither side went out of their way to turn the issue into a confrontation. Essentially, the Mughals recognized the Ottoman Caliphate, but not its universality.

The Ottomans did not respond or acknowledge this assertion (though they were obviously aware and annoyed by it), so there remained a kind of diplomatic struggle, manifested by subtleties of diplomatic language and occasional confrontations. Much of the time, the two Courts simply tried to avoid the stickier issues-- Akbar for example mostly ignored the Ottomans, and to a limited degree facilitated a diplomatic alliance against them. This explains what your Aurangzeb anecdote about-- it is correct and the implications are what you suggest; the relations between the Ottoman and Mughal courts looked something like that of two rival European empires, not enemies, but contesting some of the same terrain.

The precise nature of the Mughal-Ottoman relationship varied over time; some Mughal Emperors like Babur were very impressed with and close to the Turks (the Chagatai Turkic language was his mother tongue, and it's in that somewhat Persian influenced Turkish language that he wrote his much acclaimed memoirs, the Baburnama), while others were less interested in the Turkish cultural universe. All involved were jealous of their power, and the Mughal Emperors were regarded as "amir al-mu'minin" (Commanders of the Faithful) in their domain, a title which in the Sunni view belongs to the Caliph

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Makka and Madina were the jewels in the Ottomans' crown and their suzerainty over these holy cities had considerably bolstered their claim to khilafa. On this ground 'Abdullah Ibn Zubayr had exacted supremacy over Mu'awiya during the early days of the Umayyad rule. Later the 'Abbasids claimed primacy over the Umayyads of Spain and the Fatimids of Egypt on the same premise. On the other hand, Humayun's claim to khilafa in India was probably based on Jalal al-Din Dawwani's (d. 1501) theory that the monarch who governs with justice and directs the affairs of the state in accordance with the Shari'a was entitled to hold the dignity of the khilafa. For a detailed analysis of the Ottoman claim to universal khilafa and the Mughal response to this claim see

N. R. Farooqi, Mughal-Ottoman Relations: A Study of Political and Diplomatic Relations between Mughal India and the Ottoman Empire 1556-1748 (Delhi, 1989), 173-221.

see: "SIX OTTOMAN DOCUMENTS ON MUGHAL-OTTOMAN RELATIONS DURING THE REIGN OF AKBAR"