"two-factor authentication" on All Devices

Can I turn off two-factor authentication after I’ve turned it on?

If you already use two-factor authentication, you can no longer turn it off. Certain features in the latest versions of iOS and macOS require this extra level of security, which is designed to protect your information. If you recently updated your account, you can unenroll within two weeks of enrollment. Just open your enrollment confirmation email and click the link to return to your previous security settings. Keep in mind, this makes your account less secure and means that you can't use features that require higher security.

Source of above Two-factor authentication for Apple ID - Apple Support

Instead, it's a number that keeps changing before I can even enter it!

I've never seen the number change. How is it changing?

Also, you should only see it when logging into your iCloud account for the first time on that device. There are times when it will lose confidence in your device and ask again, but it isn't often.

sorry, I need to disagree. I get it once or twice a week - every time there is an update for the OS, iOS or the browser - got it this morning after updating to 15.0.2 on iPad) asking me to trust this browser. Mind you - no changes here: same device, same browser. This has been happening since I allowed 2FA. And, unless you react, the code will change.

I rarely get a request even after updates. Are you running through some sort of a proxy or a VPN?

You will get it if you bypass their new auth scheme such that you have to enter a username and password.

No I am not running a VPN or proxy.

What new scheme are you referring to? I’never needed a username or password for either Safari or FF. Use Safari on iPads (on latest iOS) and FF on iMac (on 10.13.6 purposely). Do not use Big Sur, something else on my test MBA. Doesn’t seem to matter which device or OS or iOS - I mostly get “trust this browser?” and a verification code regularly after updates and sometimes for no apparent reason.

Do not use Big Sur

Well, it's a Big Sur forum. I don't know what Catalina uses as I haven't used it in almost two years, now. I only occasionally jumped back to it to check on questions posted here.

In Big Sur, you get an authentication popup sheet asking either for TouchID, FaceID (iOS), or user account password. It then sends the authentication. If I cancel out of that and enter AppleID username and password, I get the two-factor authentication screen. If Catalina doesn't implement that authentication scheme, then I would suppose you would get the two-factor authentication quite regularly.

Ahh, thanks for that info; no, I am not on Catalina either - by choice I am still on High Sierra on the iMac (I am not about to spend more than $1,000 for some software that I only use occasionally now) and I did not like BS when I tested it. In any case, it makes no difference what version I am on: High Sierra on iMac, iOS 15 on iPads, or (beyond BS) on MBA - I get it on all of them. And, when I was running BS, I have never been asked for Touch ID since I do not have a Mac that has the feature. I sign in when booting up and that is it (except when I need to sign in here).

Oddly, it does not appear to. I have to log into the iPad unless touch or face id is already active. That is fine except when I do an I then get prompted for the verification code. It is either trust the browser or “a new device is being used” which is utter nonsense.

iOS 15 should be providing the same authentication mechanism as Big Sur. At least it does for me.