The first two screen shots are when private browsing is enabled , the trusted devices are verified , but its from a single Mac in which no other devices iPhone / iPad are synced .

It prompts to enter admin password , from this forum website or from www.icloud.com

When private browsing is disabled from safari preferences and set as new window , and again signed in with credentials at www.icloud.com the same admin password is required , if I click on cancel an empty field appears , type Apple ID the password is automatically filled up as iCloud Keychain was set up and in safari preferences > password section the Apple ID address and its password is saved previously ( autofill feature had already filled up the password in keychain access ) .
After that a map window appears > click on don't allow > don't change Apple ID password > the verification field appears , the code is not displayed as Mac is not synced with iPhone , click on use iPhone , the numbers are shown click on desired one the code goes to it .
So my opinion is even you clear all history and website data or if its connected with cookies the sign in method works in its own way

When you sign in on the web, you can choose to trust your browser, so you won’t be asked for a verification code the next time you sign in from that computer.
This feature works with latest models of Mac and with latest versions .