Why am I continually being asked to give verification code?

On my iMac, which never leaves its desk or the internet connection it lives on, every time I want to log on to my iCloud account or connect to Apple Discussions I have to get a verification code - the message says that my Apple ID is being used to sign in to a new device, which is nonsense, and asking whether I will allow it.  This message comes up on all my devices (e.g. my iPad) which may be in use by another family member, resulting in disruption of their work.  I thought that once I had done this once, I would never have to do it again - just use my password - but no, it happens every time, even when a session times out and I have to sign back on.

Apple's document "Two-factor authentication for Apple ID" includes this statement "Once signed in, you won’t be asked for a verification code on that device again unless you sign out completely, erase the device, or need to change your password for security reasons. 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."

I click the button which comes up asking whether I trust my browser, to say 'Yes". Shouldn't that stop this incessant asking for verification codes? What might I be doing wrong?

Thanks,

Paul.