Why does www.icloud.com always ask for verification?

There is a difference between a trusted device and a trusted phone number.

You can check for trusted decides and numbers at appleid.apple.com.

See >>> Two-factor authentication for Apple ID - Apple Support

I've experimented with different browsers trying to solve this, but mainly it is Chrome.

Some browsers may not "respect" the cookie for this. Safari does.

When I told iCloud to text the verification code to me, the text went to all three devices.

As an SMS, not a notification., Only your actual trusted devices will get the code :first"

Because you have Two Factor Authentication on.

The OP is aware of that.

The question is about not being able to have the browser save that for his next sign in. And why his codes have to be requested a second time.

I have asked this screenshot be removed as it has personal data on it that you should not post in a public space. Using your posted email address and date of birth someone may be able to compromise your account.

You are looking in your Mac's preferences. You need to sign in to appleid.apple.com as I posted.

You can only see your trusted devices at that account website.

What is your trusted device?

Which browser?

Nobody can answer your questions without more detail.

Hello LAC. I reported it a minute ago, it appears your report may not have registered and it was still there. Apologies for the intervention.

No harm done. Hopefully it is redacted shortly.

Because you have Two Factor Authentication on. To turn it off, sign into your Apple ID online, and click security, then turn Two Factor Authentication off.