There is a key difference between the slavery of Africans in the Christian lands and the ones in Muslim lands. In both cases slaves would be converted to the master's religion. Christian slave societies could accept Christian slaves, however Islam does not allow for enslaving Muslims (though that rule was broken any time enterprising slavers could get away with it, it was in fact possible to process legally if you could prove having been Muslim before being enslaved and thus effect your release), regardless of how many loopholes they used to justify for keeping people slaves, any children born would be free, no matter what. Slavery is thus not a hereditary condition in the Muslim world [in theory]. That also means there's no incentive to ensure the population of slaves is self-replicating because only "new" people can be enslaved. The opposite in many cases, as a slaveowner having children with a slave produced free people with legal inheritance rights. Basically you don't really want a self-replicating slave ethnicity because it cannot exist. It was also much easier to buy your freedom in the Muslim slave tradition and there was a great deal of pressure to free slaves when the master died as a pious act (your slaves were muslims after all). Society could also much more easily cope with the concept of free(d) blacks and integrate and absorb them back into society. Free black Muslims were not a problem for society as a whole.

The way slaves were used was also somewhat different. The Muslim world only used in a very limited capacity the type of plantation economy with a large concentrated slave population which became the "American" version. Early attempts actually led to massive issues with slave uprising and a short-lived slave state in southern Iraq IIRC. The only other really systematic agricultural plantation slave economy IIRC right now was at Zanzibar but that also happened more towards he end of the whole "Ye Olde Tyme" type slavery. The more usual type was domestic work and the ever so common slave soldiery.

Now contrast this with the "American" slavery (North, South and Carribean) of Black Africans which was a hereditary slavery ultimately based on race. One intended purpose was to create a permanent self-replicating black slave population (because buying new ones is expensive and also becomes harder, i.e. the supply is being stopped). The brutality of slavery (hard work, illness, active resistance to pregnancy etc) however to some degree prevented this, though in the US Southern states they did basically get to this point. Mostly by improving living conditions somewhat. None of the slave societies in the Western hemisphere really liked the idea of free(d) blacks and while it could be done usually massive hurdles were erected and anyone managing to clear them absolutely could not integrate into society as a normal person. Their very existence created friction to the system as a whole causing problematic questions and ideas to be raised. Every effort was made to plug the leaks out of hereditary race based slavery.

Basically the "Arab slave trade" did not produce significant demographically important populations of blacks in concentrations that society at large couldn't absorb. Whereas the "Atlantic slave trade" effectively had as a goal to produce exactly that, a black slave population that under no circumstance could be absorbed into wider society, even past it's eventual abolition.

When the British (and it was mostly the British) eventually managed to basically end the African slave trade in the classical form there was no "new blood" infused into the "arab slave trade" and the less demographically prevalent slave populations merged into respective societies. In the Western hemisphere black slavery could continue under it's own power for longer with much more solid separations between races even afterwards.

EDIT:

I wanted to check some things so I went back and had a flick through the sources I have access to where I did interestingly find figures of 15% and 25% slave populations in the areas along the Persian Gulf in the latter 1800s (the areas of modern Kuwait and UAE were the examples reports were from). I understand the OP's confusion and the best answer I can give is that the slavery doesn't seem to have resulted in populations that grew at the same rates as the majority populations in the 1900s. The "arab slave trade" in one way could be said to be more demographically varied than it's transatlantic counterpart.

Needless to say this more of general overview of a complex issues. It took Dick Harrison 3 volumes and close to 2000 pages to cover it so a lot of different circumstances exist.