Can a society truly be secular if political positions are allocated to certain religious groups?

No, and it’s a mistake to imagine that political ideology isn’t an analogue of religion in the 21st century. Critical pedagogy for instance has everything shy of an anthropomorphic godhead or pantheon that typifies a religion: saints and sinners, the righteous and the heretical, high priests and acolytes, organizational power structures that largely dictate dogma, the profound rejection of empiricism, etc.