It was not a claim, the Muslim states submitted by way of minting coins in their image and paying them tribute. Only the Kwarezmshahs were resistant to this custom and they were defeated by the Abbasids in war and later the Abbasids supported the Mongols in destroying the Kwarezmshah state. Likewise, the Mongols perceived the Abbasids to be the foremost power in the Islamic world, hence why they addressed the Caliph as their equal, yet referred to the Kwarezmshahs as their inferiors. Abbasid powers were also strong enough to be referred to by the Delhi Sultanates as their liege lords. Caliphal legitimacy and rulers of Islamic lands not submitted to the Abbasid house, lost their throne or built their legitimacy upon factors other than submission and ultimately under a non-Islamic motif, such as the Ottomans, the other Turkic Beylicks, the Almohads, the Timuro-Mughals, etc etc etc....
Once more, the Bulgars have no reason to convert. They are strong and powerful, the Umayyads are unlikely to conquer them so easily and should they, the Turkic Bulgars flee westward or northward and seek aid from the Khazars, who in otl, penetrated deep into Umayyad territory before forcing the Umayyads into a disastrous Steppe campaign that gained only a lip service Khazar vassalage which the Khazar broke the next year.
Bulgars would not view the Umayyads as advanced. The steppe folk such as the Celestial Turks, with whom the Bulgars were part of, viewed the sedentary peoples as prey and opportunities to loot, pillage and capture from them tribute. To them the notion of what they have economically is only interesting insofar as they can be taken from. The Umayyads understood this, hence why they had no pretense of friendly relations with these steppe people, instead attacking them at every chance they could in the east and north; there was no assumption that these peoples could be converted by any means other than the sword.
Thus, just as no Turkic group on the Islamic east ,convert, so too would be the same in the northwest. The Umayyads and Abbasids had been ascendant in the east in ways similar to what would be the case had they taken Constantinople, yet the Karluks, Tibetans, Neo-Sogdians, Neo-Scythians, Uyghur, Qhara-Qhitan, etc etc, did not convert to Islam, for they understood Islam as part of a wider socio-imperial complex. Only whence the Turks began entering Islam through slave trading, adoptions and general forcible application of Turkic horsemen into Abbasid government positions.
well sir you condvinced me , good talk mate