No one would ask on an application form what the applicant's sexual orientation, religion, race is etc...so why would you ask about ESA? 

The Fair Housing Act is what applies to privately owned property (with the exceptions Bruce listed above) and it forbids discrimination against prospective tenants because of their disability as well as race, religion, sex (including gender identity and sexual orientation), skin color, familial status, and national origin. 

I swear on all the religions that they dont.

Of course we have people who are unhappy with us (like every other business, non-profit, politician, author, country, fictional character, book, religion, idea.....)

Property owners/managers are subject to the Federal Fair Housing Act, which prohibits "any preference, limitation, or discrimination because of race, color, religion, sex, handicap, familial status, or national origin, or intention to make such preference, limitation or discrimination."

.§2-2 The Property Manager shall not deny service to any person due to race, color, religion, sex, handicap, familial status, national origin, sexual orientation, or gender identity.Article 3: RESPONSIBILITY TO CLIENTWhen working in a disclosed relationship with a client, the Property Manager shall protect the client’s best interest at all times.STANDARDS OF PROFESSIONALISM§3-1 The Property Manager shall use written agreements, and written extensions, if required, outlining all responsibilities and fees, if any.

And Fair Housing NC lists protected classes as "race, color, religion, national origin, sex, familial status, and disability".I think the person you spoke to at the PM company misunderstood the law or miscommunicated it.

The race, religion or sex of the tenants could change overnight?

Then I "got religion" concerning title when I score a $500K equity profit on a deal in 1991 because of a superior title information.

Had a feel of manipulation and indoctrination into, I hesitate to say cult or well known 'religion' famous for this but they are out to get max hooks into you, get max money up front, and keep tapping you as you are cast into the 'sunk-cost' fallacy.