Airplane Mode used to be a true “all wireless disabled” mode, but Apple have relaxed it a bit over time, possibly because it's so frequently used and there are so many services provided by WiFi and Bluetooth now that aren't related to the Internet (e.g. connecting to AirPods or an Apple Watch). They also relaxed what turning off WiFi and Bluetooth in the Control Center (separately from Airplane Mode) do, so that things like AirDrop still work. But I think the options in the Settings app are still meant to truly turn off wireless, rather than just mostly?
Also: if Settings says you're connected to a WiFi network, but you don't see a WiFi icon at the top of the screen, I think that means there's no working Internet connection.
Also: if Settings says you're connected to a WiFi network, but you don't see a WiFi icon at the top of the screen, I think that means there's no working Internet connection.