How do I get my iPhone 15 to save passwords

See the following from Set up iCloud Keychain - Apple Support

Turn on iCloud Keychain on your iPhone, iPad, or iPod touch

1. Tap Settings, tap [your name], then choose iCloud.
2. Tap Passwords and Keychain.*
3. Turn on iCloud Keychain. You might be asked for your passcode or Apple ID password. 
4. 

Find your saved passwords and passkeys on your iPhone or iPad

One or all of the following may help:

Set up iCloud Keychain - Apple Support

https://www.macrumors.com/how-to/icloud-keychain-iphone-ipad/

https://www.macworld.com/article/331902/apple-id-icloud-keychain-passwords-notes-safari-chrome-iphone-ipad-mac-ios-macos.html