Meditation doesn't come from religion. Quite the opposite. In Hinduism, there is the concept of "ahamkar," which could be roughly translated as "self-action." It is considered false because awareness/nirvana/void, is not exclusive to any person or tradition and everything happens at once and is over just as soon as it happens, so Hinduism says a symbiosis with God (which in Hinduism pretty much means the universe) comes from not identifying oneself with actions, practices, beliefs, etc. The religious texts and practices are considered roads to nirvana rather than beliefs. Those are considered products of the mind, rather than the mind's state being products of the beliefs. When one detaches from ahamkar, it is as if they ride their body and mind, rather than being their body and mind and all its beliefs and practices.
This is more pronounced in Zen. In Zen, there is the "I want to hear YOU scream" thing, where they get someone to scream in a way that is not premeditated, but natural and in flow with the present moment. Unlike a more Western view where beliefs shape the mind and action, Zen holds that the mind is only shaped by beliefs when it is not aware enough of its timeless awareness.
In advaita ("not two"), Maharaj Nisargadatta says that it doesn't matter what religious practices one uses to find nirvana. You can do it through Hinduism, Buddhism, yoga, or even without any of them. Nirvana is in humanity regardless of beliefs, culture, etc.
There are practitioners of Buddhism, Zen, and Hinduism attach themselves to meditation like Christians to the bible, but to reach nirvana, meditation is not a requirement. Though it certainly can help. Some Buddhists will say "only this type of meditation will bring enlightenment," but they're not being true to Buddhism's absence of attachment from beliefs. It can be weird for Westerners to wrap their head around a "religion" of no religion, but that's the case. Meditation, like mathematics, is a discovery that exists outside the context of culture.
I wouldn't say meditation gives an alternate experience of reality so much as an amplified ordinariness. Though it is a bit different experiencing days and weeks and months as a single moment where everything happens spontaneously.
Meditation doesn't give knowledge of consciousness one doesn't already have, but detaches from outside obscurities so that the world can touch the deepest parts of one's psyche more easily. Everything makes more sense because all of one's life is a single moment where everything is connected. But the connections are wordless.
You'll notice if you read a sentence and don't say the words in your head that the meaning is still there even if you close your eyes and keep your mind silent. It's like words bounce of a more base state where "making sense" happens. Meditation can expand this to make it constant. It can be a bit disorienting at first, because memories lead to action without explicit expression in the mind. It's like knowing a scene in a movie and experiencing it without watching it, sort of a mental blindsight. Putting stuff into words can be frustrating because the mind is freed up to take in more sensory nuance when it's not describing anything.
It's like you're a photographer and you're used to cropping your photos for websites so you take them cropped (I don't think this is possible, but for the sake of argument it is), but then you stop cropping them and see more, so then when it comes cropping time, it's like, "but how? I'm missing so many things."
Edit: To illustrate the East/West disparity a bit more, I think it was Thich Nhat Hanh who said "existence and nonexistence have no place in discussions about reality." Certain sensory phenomena come into and out of being all the time, so one must look beyond them.
My perspective is a mix of East and West. I still use concept and use beliefs (rather than have them). I don't agree with the demonization of desire that some Buddhists have. I think the best desires form spontaneously and enrich one's life, and steer clear of desires that are about avoiding pain or taking the easy way out. I enjoy the Western progressive mindset which some Easterners find opposes contentment, but I think I combine both. I seek higher pleasures and rewards while being content where I am. I ride waves up and down and try to make them go up, rather than just stay where I am. I think it is foolish how some people attach to meditation, since life is an infinite mystery about more than just awareness. The universe has great variety and change, so why miss it? I also don't believe concept should be eliminated altogether, since it has been used to tap into secrets of the universe that give us great toys like the internet. In fact, rather than calling all concept false altogether, I consider it another form of perception not to be taken as the only one. Concepts like mathematics have observable results, so how is concept altogether false?