This is somewhat correct. Western philosophy (and even the natural sciences) operate under guises of Western religion for very long. Even iconoclasts like Nietzsche and Marx do so. Religion and faith subtly intertwine with philosophical discourse.
This is the same for a lot of cultures. I'm not familiar with a lot, but I am trained in Chinese philosophy. It operates under the metaphysics of their religion. Confucianism, Daoism and Buddhism are all contemporary religions. But the idea of 'religion' is vastly different in the East - as you might notice.
As for the second part, I haven't come across any terms in Chinese philosophy that are immediately translatable to 'reason', 'evidence', 'morality' or 'religion'. These are all terms that were imported into Chinese culture in the late 19th century following Western imperialism. As such, you won't see debates about God, religion, reason or evidence in classical Chinese philosophy.
That's why in experiencing and studying philosophies and histories of a different culture, I come to realise that a lot of the concepts we employ in discourse aren't actually 'objective', 'impersonal' or 'universal'. So what you think and say as 'evidence' and 'religion' is - largely - conceptually unintelligible if I don't subscribe to the interpretations of those concepts.
I have a lot to say, actually. But I hope this answers you. Thanks for engaging :)