Two different things get called "Wi-Fi search history": what the router saw, and what specific devices did while on that network. Both are accessible, but not from the same place.
1. The router's DNS log. Open your router admin page (usually 192.168.0.1 or 192.168.1.1, credentials on the sticker), find Logs, DNS Log or Traffic Monitor. Most consumer routers keep a rolling buffer of a few hundred to a few thousand entries, and many only log if you enable it. What you get is domains — google.com, youtube.com — not full URLs, and HTTPS means the path is not visible. If the router uses the ISP's DNS, the log may be empty because resolution happened upstream.
2. Parental-control features on the router. Many modern routers (Asus, Netgear, Eero, TP-Link Deco) have per-device filtering and history in their app. This is more readable than raw DNS logs and lets you block categories per device. It is the practical option for a family network.
3. The device itself. Browser history on the phone or laptop is the accurate record, and that is what matters if you administer the device. On Android with Family Link you can see per-app usage but not URLs; on Windows or Mac with a monitoring agent you get full browsing history.
What you cannot do: retrieve browsing history from a Wi-Fi network you do not control — a café, a school, a neighbour's. And you cannot see another adult's browsing from your router without crossing into surveillance territory; on a shared home network, that is a conversation, not a configuration.
For legitimate household visibility, the cleanest stack is router-level filtering for the whole home plus device-level history for the machines you administer. Ultimate Phone Spy covers browsing history on Android, Windows and Mac (web history, plans), which is more reliable than DNS logs because it records the URL the browser actually loaded.