Well the "Air" part is for air mattress, which is no longer true, if it ever was. The name was supposed to connotate a very spartan take on a traditional bed-and-breakfast. I've never heard of anyone booking all-inclusive travel through airbnb.

In my experience, and I've stayed at a lot of them, BnBs usually have a hot sit down breakfast but it's not universal. I've certainly stayed at BnBs where the host is politely but firmly clear they're not there to entertain you and there will be coffee out in the morning and feel free to grab some yogurt from the fridge. While there is usually fruit and muffins that you can grab for lunch, they don't usually provide a brownbag lunch as such and rarely dinner.