I have a colleague that does this with their voicemail. Almost always forgets to change the message once they’re back in so if you call, say, on Thursday, June 3, and you get their voicemail, you will hear how the person is out of the office from May 24 to 26 and who to call while the person is away.
I wouldn’t be offended or consider saying anything to anyone who included this in their out of office message, but even as someone who is in a religion that forbids use of electronics on most holidays, I still think this message is 1) TMI; 2) doesn’t convey what it needs to convey unless you are explaining that you don’t use electronics during holidays, in which case you can just include that you won’t be checking email without including the religious explanation; 3) would come across to me as inclusion of personal information I don’t need, which would therefore strike an unprofessional tone,and I wouldn’t understand why you felt the need to include that info. How about “I’m out of the office without access to internet or email until (date). If you need assistance before this date, please contact…” Like I said, I think your colleagues are overreacting, but in general I would advise to leave all personal information out of your auto-reply — vacation, medical leave, religious observance, etc. — people do not need to know why you are out.

The OOO definitely has those two pieces. But it could be 1 of 4 people who handle things when I’m out (depending on what it is) and they may not know at all that the requestor had reached out to me first / forget to cc me. So, I’d see this as me adding to my OOO “if you contact person X, please keep me cc’ed on the message you send to person x”?
I don’t. They just have to wait. The people in my office that could have a legitimate emergency that requires my input have my cell number, and they’d just call me.