I don't know why the existence of aliens would rattle anyone's belief. Unless you can find a way to take away their threat of dying, they're going to believe in God.

The irrational will always find ways to remain irrational.

I think the discovery of intelligent life outside earth will cause existential crisis to a lot of people, and some of them would turn to (new?) religions to cope.

If the discovery of life on other planets is anything like the discovery of life on other continents, then yes, we will force them to change their religion (and economic model).

Oh, is that not the question?

Well, it's a suitable answer anyway. No way we'd change our beliefs to accommodate aliens when we can change the aliens to accommodate our beliefs.

Like everything else, it depends on which religion you ask. Some will be fine with it, while others will be frothing at the mouth in denial, to convert them, or call for full on extermination. Just ask yourself "how does X group treat the humans they hate the most" and you'll have your answer.

Really depends on the religion and the kind of life discovered. No religion would have any issue with life on other planets in general, as long as it's not provably intelligent life.

If it is provably intelligent life, some branches of Judaism, Christianity, Islam and Mormonism (in order of age) would get into a crisis of belief while other branches would readily adapt. There likely would spring up many sects, some of which would claim that intelligent life to be angels/demons. There would be theological debates about whether these beings have their own original sin, etc.

Meanwhile, most other religions would simply not care all that much. Ancestor worship literally wouldn't give a shit. Religions with reincarnation would simply expand their notions of what you can be reincarnated as. Many local pagan religions have creation myths that apply only to their local region. If those creation myths survived contact with colonizers and the knowledge that the world is larger than the area they cover, the knowledge of aliens existing wouldn't make any more of a difference.

Also, I wouldn't be surprised if several new alien-based religions sprang up.

Some fandoms are basically religions, especially sport club fandoms. They wouldn't give a shit.

Lastly, it would be a boost to militant/proselytizing atheism/materialism, which also are like religions to some people.

The real chaos would happen if all the recent UFO stuff turns out to be true. Aliens existing RIGHT HERE ON EARTH could quite possibly cause changes in one or more major religions, but maybe not immediately.