One recipient said she was "truly appalled" at the change.
Dark blue passports resembling the pre-EU British design are due to be issued from the end of the year.
Susan Hindle Barone, who received her new passport on Friday, told the Press Association she thought the design should not change for as long as the UK remains an EU member.
She said: "I was just surprised - we're still members of the EU. I was surprised they've made the change when we haven't left, and it's a tangible mark of something which I believe to be completely futile.
"What do we gain by leaving? There's certainly a whole lot we lose."
A change in the design of the UK passport has proved a rallying point for Brexit supporters, with former UKIP leader Nigel Farage describing the 2017 decision to bring back the dark blue design as "Brexmas".
The decision to remove the "European Union" label was made in the expectation that the UK would be leaving the EU at the end of last month, as scheduled.
A Home Office spokeswoman said that "in order to use leftover stock and achieve best value to the taxpayer", passports that include the words "European Union" will continue to be issued for "a short period".
She said: "There will be no difference for British citizens whether they are using a passport that includes the words European Union, or a passport that does not. Both designs will be equally valid for travel."
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