No, I wouldn’t mention the competition in your listing, but rather highlight what you offer that they don’t, washer/dryer, kitchen, pets, yard, etc.
as for pets, we offer crates, beds, furniture covers, bowls/placemats, poop bags. Not everyone uses everything, some bring their own beds, crates, etc. but we recently downsized our primary residence and the airbnb is our second home so we had extra stuff we moved into it.
by providing these items it’s sets a precedent that we require the use of furniture covers, crates, placemats, poop bags. So far no issue with pets, just issues with people.