I’m simply repeating what Barnier has stated, not what Keir Starmer’s towel boy, the quivering, clueless Barry Gardiner says.
It’s very clear: as part of a customs union, the UK loses the ability to set its own trade policy (wrt goods) - it cannot set its own tariffs in bilateral trade deals with other countries and must accept whatever external tariff is set by the EU.
It goes without saying that businesses trading in the customs union have to comply with EU regulations. But that’s true for any business, whether American, Chinese or Venezuelan that wants to export to the EU and would continue to be true if the UK crashed out without a deal onto WTO terms. If that’s your point, it’s beyond banal.