Singh is a name used by Sikhs and Hindus. I believe it means lion and it harkens back to the kshatriya caste of warriors. (Hence Marla's remark about warriors.) As Christopher said, the episode is more a garbling of various Indian cultural elements then a true representation. The episode does indicate that a lot about him isn't really known.
In story it's not necessarily unlikely that Khan was a Sikh by birth/baptism who later gave up or took on another belief system/philosophy. I haven't read the books, but they're on my Kindle for once I get caught up on the 24th Century Stuff.