No, Airbnb the company can't see your Wi-Fi history. It doesn't run the internet at the properties listed on its platform. The host can, though, because the host owns the router, and a router can log what passes through it.
What a host can see
Most consumer routers show at least a list of connected devices, their names, and how much data each one used. Some keep logs of the domains visited, and hosts using DNS filtering tools or custom router firmware can get more detail. With HTTPS, which covers most sites and apps, a host can't read your messages, passwords, or the specific pages you open. They can still see the names of the sites you connect to, since that information isn't encrypted on most connections.
What Airbnb's rules cover
Airbnb's device policy deals with cameras, not networks. Since April 30, 2024, indoor security cameras are banned in all listings, even if switched off, and outdoor cameras must be disclosed and kept away from private spaces like outdoor showers. Noise monitors are allowed indoors if disclosed, but they measure decibel levels and don't record audio. Smart speakers like Alexa are allowed, and Airbnb encourages hosts to let guests unplug them. NPR reported the camera ban as a response to longstanding guest complaints.
The policy doesn't mention router logging, so don't assume it's off-limits.
How to keep your browsing private
Use your phone's cellular data for anything sensitive, or run a VPN such as NordVPN on the host's Wi-Fi. With a VPN active, the router sees only an encrypted connection to the VPN server and the amount of data used. Also rename your device if it includes your full name, since that shows up in the host's device list.