Can an employer demand to know my religion?

Well yes, if you request a particular exemption, you need to provide some proof of your reasoning.  

Let's figure this out together.  I'm in Canada, BTW.  If you request an exemption under human rights to keep a beard due to your religion, do you have to advise which religion you practice?  Yes.  Doesn't that make sense to you?

I could claim the right to grow a beard because I'm Catholic.  Beards and Catholics have nothing in common.

You are requesting an exemption based on religion. They CAN'T grant it without knowing the religion.

I am not a lawyer, nor even live in Canada, but I can read Canadian Law. The Human Rights Act does not specifically protect a beard. He can deny that exemption. He is asking under what provision of the law that protects against discrimination the beard is required under. Certain religions, such as Sikhism, requires a beard.  He asks which religion you practice that does not allow you to shave your beard. If you are a practicing Sikh, he could research other details of the practice to confirm your claim, or explain to other people why they cannot wear a beard. Of course, if caught lying, you can be fired for that.
For example, if you state "I am a Sikh and cannot shave my beard" then his/her reply is "Where is your turban?"
The court in Ontario has already ruled that employer’s “clean shaven policy” cannot be regarded as amounting to discrimination based on sex or gender expression. So, there is religion, ethnicity or national origin or genetic characteristic. If you grow the beard to hide permanent scarring or a discoloration birthmark, he can request a verification and ask you to trim the beard to a reasonable extent. You are not required to state your religion, and your employer is not required to allow your beard. It is a standoff that you lose because you only state "Human Rights Act" and not the specific provision.
https://laws-lois.justice.gc.ca/eng/acts/h-6/page-1.html#h-256800

You don't have to shave your beard in Canada, if it's part of your religion. The Canadian charter of rights and freedoms protects citizens when it comes to race, age, religion and sexual orientation.

you might need to do that

Absolutely fecking not. You not under any obligation to provide this.