I think you have a slight misunderstanding. Muslims can absolutely have Muslim slaves. The Mamluks for whom the Ottomans' predecessor empire was named started off as, yup, Muslim slaves. The famous "singing girls" of the medieval Islamic courts from al-Andalus to Baghdad were almost all slaves. Muslim children of Muslim slaves are still slaves. (This only ends up true in rare cases; see discussions below.) The issue is that a Muslim cannot enslave another Muslim, that is, they cannot take a free Muslim and make him or her into a slave.

The janissaries were indeed typically taken from the Orthodox Christian population (until the corps became prestigious enough that Muslim families volunteered their sons freely); however, their education amounted to a forced conversion. (Of course their teachers would have maintained conversion was simply the natural result of good moral instruction and perhaps peer pressure.) For example, 16th century janissary Sokollu Mehmed Pasha was Serbian Christian by birth but eventually became grand vizier, and in his extreme wealth he founded/patronized mosques all over the Empire.