Taking Zamp's ideas and developing them. We have some great risks of stereotyping here, but here would be my thoughts:
Hinduism is known for its pluralism.
So I suggest that influence with city states is greatly increased (like Philanthropy and/or Greek UA)
Buddhism is known for encouraging peaceful states of being and building huge monuments!
So I would suggest zerofor number of cities and +20%
for wonders (like a part combination India/Egypt UA)
Judaism's greatest modern contribution may be through the arts, journalism and finance.
So I would suggest +5%increasing by +5% for each later era with better communication, and +10%
, both empire-wide.
Christianity is an evangelical religion with a strong work ethic.
So I would suggest +0.5/city population point in each city and +15% troop bonus.
Islam is an evangelical religion with a trading and science contribution.
So I would suggest +10%and +10%
empire-wide, and +15% troop bonus.
Confucianism is known for the importance it puts on social structure.
So I would suggest new social policies should be cheaper to get by 33%. (like Free Speech/Redentor)
With no adopted religion, there is more freedom for original thought.
All science buildings (libraries etc.) are more effective with all normal +50% bonuses rising to +75%, plus +25%Scientist points.
Taoism (missed by Zamp) could emphasize the acceptance of and working with natural flow.
So I suggest -50%for number of citizens, plus units have no terrain penalty in friendly territory.
Confucianism is inherently anti-change. They want to maintain social order and are extremely conservative. I'd say they should get bonuses in other areas but make it harder to pick up new social policies in my opinion.