I can't log out of my iphone without losing my passwords

Is the feature switched on? Set up iCloud Keychain. Also note:

"What happens when I turn off iCloud Keychain on a device?

When you turn off iCloud Keychain for a device, you're asked to keep or delete the passwords and credit card information that you saved. If you choose to keep the information, it isn't deleted or updated when you make changes on other devices. If you don't choose to keep the information on at least one device, your Keychain data will be deleted from your device and the iCloud servers. "

When you log out, your keychain and other information is removed as you are no longer logged into the device.

Because the passwords are saved in the keychain, which will be removed/cleared when you sign out.