Why does my phone show up on my family member’s phone as an available hot spot when my WiFi is disabled?

Disabling the Wi-Fi connection on your phone stops it connecting to any locally available Wi-Fi network, but clearly doesn't shut down its ability to use its mobile data connection to provide hot spot access to other devices that you choose to allow to connect. Turn off the hot spot or change the password to prevent unwanted connections.

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From what I can tell from these two documents, it doesn't specify that using a host device for hostpot requires wi-fi be activated on the host device. The guest device (the one trying to connect) does need wifi active

If Personal Hotspot is not working on your iPhone or iPad (Wi-Fi + Cellular) - Apple Support

How to set up a Personal Hotspot on your iPhone or iPad - Apple Support

On the device that you want to connect to, go to Settings >

Cellular > Personal Hotspot or Settings > Personal Hotspot and

make sure that it's on. Then verify the Wi-Fi password and name of the

phone. Stay on this screen until you’ve connected your other device to

the Wi-Fi network.

No mention that the host device has to have wifi active.

On the device that you want to connect, go to Settings > Wi-Fi and

look for your iPhone or iPad in the list. Then tap the Wi-Fi network to

join. If asked, enter the password for your Personal Hotspot.

This is the guest device

No idea. If you think it is a bug report it via Feedback - iPhone - Apple.

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