You were using an iMac previously , but you sold it out and did not signed out it from Mac system preferences What to do before you sell, give away, or trade in your Mac - Apple Support
Also the iMac and Mac book air were signed in with admin name and password on the same network , and same Apple ID and password .
The iMac still exists in server account see the article to remove it Check your Apple ID device list to see where you're signed in - Apple Support
Come to your Mac book air , it will not prompt to enter admin password for iMac , it is prompting to enter admin password of Mac book air .
You can click on cancel and again sign in with your Apple ID and password with account credentials in MacBook Air system preferences Sign in with your Apple ID - Apple Support
You have removed passwords from Mac safari preferences , Change Passwords preferences on Mac - Apple Support
Remove websites you selected in the list.( selected them and clicked on subtract sign ) .
however if iCloud Keychain is set up Set up iCloud Keychain - Apple Support
And other devices are synced with the Mac book air Make your passwords available on all your devices with iPhone and iCloud Keychain - Apple Support
The website password will sync with Mac book air , also in safari preferences autofill box is checked , those passwords will be filled in keychain access and you as an administrator can literally view View the information stored in a keychain on Mac - Apple Support