I don't know about the Byzantine case, but I'll try to answer the Ottoman ones. The reason why the Ottomans chose to made Crimea her vassal had to do with how the Ottomans vassalized Crimea. The Crimean Khanate was founded in the 1440s by Hajji Giray, the descendant of Genghis Khan's grandson Toka Temür. After his death, the Khanate fell into a civil war between his sons Mengli Giray and Nurdevlet. Nurdevlet, supported by the Great Horde and Genoa managed to beat Mengli Giray, who then later escaped to Caffa and sought help to the Ottomans who were also battling the Genoese. After the fall of Caffa and Azov in 1475, the Ottomans restored Mengli Giray to the throne, but as an Ottoman vassal. The ascension of Mengli Giray as an Ottoman vassal linked the Crimean Khanate and the Ottoman Empire on the basis of mutual interest, the Khanate was generally independent in terms of their foreign and internal affairs and enjoyed Ottoman protection, but they had to pay tribute and provide troops for Ottoman campaigns. Even this independence was becoming more and more limited as time goes on, where in the 17th century the influence of the khans had been severely limited by Ottoman intervention who could dismiss and appoint khans to their will, and the election of the new khan by the Tatar tribes is only for formality.

The Ottoman method of giving vassalage or suzerainty to local ruler isn't unusual for the Ottomans, who often did not remove the dynasties that had ruled in pre-Ottoman times, but instead maintained them as vassals under Ottoman suzerainty as in the case of Wallachia, Moldavia, and various Anatolian clans before their annexation

That's also kinda the case for Russia after the war of 1768-74. In the treaty of Küçük Kaynarca, The Russians didn't immediately annex the Crimean Khanate, but only gave them "independence" and appointed Şahin Giray, Khan loyal to Russia. Only after the Ottomans violated the treaty of Küçük Kaynarca by supporting Şahin Giray's rival claimant Devlet Giray then the Russians annexed Crimea.

Sources:

"The Ottoman Empire and The World Around it" by Suraiya Faroqhi

"The Encyclopedia of Ottoman Empire" by Gabor Agoston et al