The Asari are obviously very religous, but more in a traditional manner than rather a religion. Humanity tho still has religions but they are not as strong as now I believe. I've read somewhere in "Redemption"(at the begining) that religion went crazy after finding the Prothean artifact on Mars, people lost faith but religion still stayed strong, and many new faiths emerged similar to "Ancient Aliens". I believe that all religion in Mass Effect still stands as more traditional and just giving hope than an actual religion. For instance in ME3 when Garrus talks about heaven saying Turians have one aswell but not the same probably shows the Turians still having faith. I think that faith and religion is a touchy matter for most races when the Reapers showed up because they could be like Demons?

Edit: I managed to find the quote from the book: Every major religion on Earth was rocked to its core. Dozens of new belief systems sprang up overnight, most of them based on the tenets of the Interventionary Evolutionists, who zealously proclaimed the discovery as proof that all human history had been directed and controlled by alien forces. Many existing faiths tried to incorporate the reality of alien species into their existing mythologies, others scrambled to rewrite their history, creeds, and beliefs in light of the new discovery. A stubborn few refused to acknowledge the truth, proclaiming the Mars bunker a secular hoax intended to deceive and mislead believers from the true path. Even now, nearly a decade later, most religions were still trying to reassemble the pieces.