A British couple who applied for their passports on the same day received different versions - one with European Union on the cover, the other without.
The new burgundy passports were introduced from 30 March, the day after the UK was supposed to leave the EU.
Peter Brady said he was "very happy" he received one of the new passports and his partner was "unhappy" she did not.
The Home Office said some people may still receive the old version until stocks run out.
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.
Dark blue passports resembling the pre-EU British design are due to be issued from the end of the year.
Mr Brady and his partner Jan both sent off their passport renewal applications on 21 March.
His passport, which does not have any references to the European Union on the cover or inside, was printed on 1 April.
His partner's passport, which was printed on 4 April, features the EU logo on the front and the inside.
Mr Brady said he feels like he has his "identity back" as he was a great believer in the UK coming out of Europe, adding it was a "shame" his passport was not blue.
"For me to have the European Union wiped completely off my passport is good news," he said.
His partner Jan was "very unhappy" as she too wanted a UK passport without the EU on it, according to Mr Brady.
A possible reason for the difference in their passports might be that Mr Brady's came from Glasgow and his partner's came from Peterborough.
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."
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".
Not everyone is happy at receiving one of the new passports - one recipient said she was "truly appalled" at the change.
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.
TRULY APPALLED. Picked up my new passport today - my old one expires in the next couple of months. See below: Spot the difference! pic.twitter.com/R7BW9lk6I5— Susan Hindle Barone (@SpinHBarone) April 5, 2019
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."
Meanwhile others are pleased they received one of the old passports after 30 March.
Steve Rowe said: "I received my new passport this week with a start date of 1 April, happy to say it still says European Union; I think we'll still be discussing Brexit when it runs out in 2029."