My landlord was named The Fifth Frogmouth Corporation.
He from Innsmouth way?
Manhattan!
Good. At least they weren’t one of those bridge-and-tunnel eldritch horrors.
As a long-term renter in two states I can confirm that “lessor” and “lessee” are the standard terms in the actual legal document called a lease.
Pretty much a match for what’s on Wikipedia:
It’s gender-specific, sure, just as “lord” and “lady” are gender-specific terms for men and women in certain classes of English nobility - but sexist? As in prejudiced or discriminatory? I don’t think it’s any more sexist than “king/queen,” “sir/madame,” or any of the other male/female corollary terms that are out there.
I’m surprised that women want, or are even willing, to drop female-gendered titles in favor of male-gendered titles. Flipping this around, I don’t imagine many landlords would be willing to be called a landlady.
In (local) French we don’t have a direct translation for “landlord”, we usually use propriétaire (“owner”) instead. Which sounds factual and harmless… but you end up with sentences like “I have to ask my owner” or “The Tremblays were thrown out by their owner”.
YMMV, of course. But who cares what the gender of the property owner is? It shouldn’t matter. And if it doesn’t matter, then everyone is a landlord, or an actor, or a flight attendant (v steward and stewardess).
I’ve never liked being referred to as a “lady.” At best, it implies that I should be bound by certain strictures of refined behavior (“you need to be more ladylike!”). At worst, it has slightly sullen, aggressive undertones: “Hey lady! Watch it!”
Like I said, YMMV. If you have positive experiences in your life being referred to as a “lady,” then I can see why the term seems inoffensive to you.
I wouldn’t mind that, but it’s going to bother some people to use a traditionally gender-suggesting term universally—like “everyone is a chairman” or “everyone is a mailman” or “everyone is a cowboy.”
Everyone is a human.
We’ll know we’ve finally arrived when a Queen Charles sits on the throne.
More likely that a similar but non-gendered term will substitute, like a chair (that’s what we call the head of our board, we just leave off “man”), mail carrier, flight attendant, etc.
My late first wife was strongly anti-sexist. But was a word maven with a humorous streak. She said:
The post office has recognized they have a terminology problem. Their enlightened management has decreed that henceforth …
The person person will carry all the person in their person truck and put your personal person into your person box.
So there! 
I’m not - historically, categories labeled female get less respect, and the way language is constructed (at least most Indo-European languages) the male form is used when 1) the noun(s) are all male, 2) the speaker is uncertain if the noun(s) are all male, or 3) it’s a mixed group of male and female, so in essence the default is to the male form and the female is a special case/exception.
The mere fact that, as you point out, few if any men would want to be referred as “landlady” further emphasizes that it is the male form that has respect, not the female.
In legal usage, “landlord” is, and always has been, a gender-neutral term. If an owner of land grants exclusive possession of the land to another person in return for rent or other compensation they are “landlord” and “tenant” respectively, regardless of the sex of either of them. “Lessor” and “lessee” are alternative terms, but they are wider terms, used in relation to leases of any property, not just land — vehicles, machinery, whatever.
“Landlady” describes a woman who runs a guest-house, pub or other facility, providing short-term accommodation and other forms of hospitality to “guests” or “customers”, not “tenants”. But that’s not really a legal usage, more a common usage.
My (soon-to-be-ex) landlord is a woman. It never occurred to me to call her a landlady.
Ouch.
If people are going to start referring to me as Lord Geek, I should probably get some fancier clothes. Currently my fanciest clothes are literally jeans and a T-shirt. Wearing my newer sneakers is what I consider to be “dressing up”.
My tenants just call me by my first name. I guess I’m not doing this pompous thing very well. The township, utility companies, and my insurance company all insist on calling me Landlord though.
I can also confirm that I have spent very little time a-leaping.
“Landlord” and “Tenant” are more common around here (southern PA). FWIW, my standard lease uses “Lessor” and “Tenant”. It was drawn up by a lawyer. I didn’t choose the terms.
It does not surprise me that there are regional differences in usage in the US.