I would be very surprised if any beach house, or any vacation rental house at all, did not provide bed linens and towels, plus at least basic kitchenware. The first place you wrote to sounds bizarre (and from the owner's response, I'd run away fast).
By the way, I'm not sure you are familiar with the individual character of the various beach towns in NJ. Stone Harbor is higher-end and family-oriented to the point of extreme boredom. Wildwood, on the other hand, is more on the "Jersey Shore" (the TV show) end of things--a younger, trashier, partying kind of place. (Although I am not much of a party type myself, I actually prefer Wildwood to Stone Harbor/Avalon because at least you can go out at night and get a drink, vs. the latter two towns where the sidewalks are more or less rolled up at 7:30 PM.)
The condo you linked looks very nice and the good reviews are a very good sign. It's also a good sign that she doesn't rent to people under 22, although I'm not quite sure how she would enforce that since Airbnb doesn't keep track of birthdates.
As for soap and shampoo, the Airbnb listing should specify whether they're provided, and if it doesn't, just ask the host.
Last edited by travelmad478; Jun 4, 2017 at 7:14 pm