One person who received the passport said was "truly appalled" by the removal of the EU.
The virtual dog has already had more than 4,000 "conversations" since its launch previous year and is designed as an approachable first step for people who think they may need more professional help.
Ahead of an European Union summit on Wednesday, she was forced to ask them for another extension, until June 30, to prevent Britain departing with no deal at the end of next week.
But the move has angered those applying for new passports who were hoping to hold on to an emblem of European Union membership.
The Home Office has warned the residents will not be issued passports that include the words European Union after the United Kingdom has left the EU - but when that is likely to be remains uncertain.
Mr Brady told the BBC: "For me to have the European Union wiped completely off my passport is good news", before adding his wife was very disappointed to get an old version. You will not be able to choose whether you get a passport that includes the words European Union, or a passport that does not.
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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".
It said the discrepancy was due to an attempt to save public money and insisted both designs would be "equally valid for travel".
Dark blue passports are scheduled to be issued from the end of this year.
The new production contract is to begin in October 2019, with the passports now being issued without reference to the European Union on them still in the burgundy colour.
British passports have become ensnared in the country's Brexit divisions after the government announced in 2017 it would return to traditional blue passports "to restore national identity". Britain switched to burgundy in 1988 in line with most other countries' passports in what was then the European Community.
The new production contract is to begin in October 2019, with passports now being issued in the burgundy colour.