Though there are several people who have rewritten very beautiful and moving religious lyrics to this music and sung them.... that is NOT what Bon Jovi sang. He sang Cohen's original version.
Here are Cohen's lyrics (in case you have never heard them):
Now I've heard there was a secret chord
That David played, and it pleased the Lord
But you don't really care for music, do you?
It goes like this
The fourth, the fifth
The minor fall, the major lift
The baffled king composing Hallelujah
Hallelujah
Hallelujah
Hallelujah
Hallelujah
Your faith was strong but you needed proof
You saw her bathing on the roof
Her beauty and the moonlight overthrew her
She tied you To a kitchen chair
She broke your throne, and she cut your hair
And from your lips she drew the Hallelujah
Hallelujah, Hallelujah
Hallelujah, Hallelujah
You say I took the name in vain
I don't even know the name
But if I did, well really, what's it to you?
There's a blaze of light
In every word
It doesn't matter which you heard
The holy or the broken Hallelujah
You are correct.
Here are some plausible interpretations:
https://www.lyricinterpretations.com/leonard-cohen/hallelujah
It’s a song about David and Bathsheba, more or less, with a little muddled imagery from the story of Samson and Delilah tossed in.
If you read the “broken Hallelujah” as part of David’s famous repentance (see the 50th Psalm — you westerners who use the Masoretic numbering will find it at 51st — and the account in 2nd Samuel), then it’s a religious song. If you read it as being David still mired in sin “having joined murder to adultery” as Orthodox hymnography puts it, then it’s irreligious.