alt Hacker NewsI think they don't approve of the Crown as an authority even if they agree it's lawfully established?
It's more like an atheist refusing to swear an oath before God in a courtroom: even if you agree that the law says you must do so, you might still not want to give God that recognition. But worse, because God might also be the defendant and the judge in this case, and you have to swear not only that He might witness your testimony but also that you pledge allegiance to Him, so swearing that oath really impairs your ability to participate in a fair trial.
For oath swearing it turns out atheists weren't why we fixed that. Some of the Christians also refuse to swear oaths. For Quakers obviously God exists - they're Christians, but a mere court of law here on Earth is no reason to go around swearing when they believe God has explicitly ordered them never to do that. They'd need orders from God, not some judge.
So for them rather than for atheists England made it possible to Affirm that you're not lying. This will work in Parliament, and it's pretty routine these days that a new member is like nope, no swearing for me, I can promise I'm not lying but I never swear or I won't swear to God.
However, Parliament does require allegiance to the King because this is a constitutional monarchy, if they wanted to be a Republic they'd get rid of the King as a group, that's not up to you as an individual member. Not much notice is taken of how much you seem to mean it about allegiance to the King, because after all plenty of members are known to hold Republican sentiments, but you are required to say the words unlike the stuff about swearing which is optional.