Does the choice of religion (not beliefs) have any impact?

I just had the chance to found my first religion, and am offered with a list of religions to choose from, such as Buddhism, Judaism etc. I understand that I then choose a founder and follower belief and they have an impact on the game, but does the religion choice itself has any impact?

In other words, is the choice between Shinto and Christianity purely cosmetic, or does it actually affect something?

The choice is purely cosmetic. BTW, You can rename them. I suspect you can even rename christianity "Shinto" if you want. The game is just asking what name and icon you prefer. The beliefs are the actual effects of the religion, as you thought.