There will be an independent review after the NHS transplant authority confirmed 21 cases in which the wrong organs may have been taken from donors.
As many as 800,000 people on the UK donor register may have had their preferences about which organs they wished to donate recorded incorrectly.
Potential donations from these people have been temporarily put on hold.
Health Secretary Andy Burnham said he regretted the error but it should not stop people from donating.
The British Medical Association warned public confidence could be damaged.
NHS Blood and Transplant apologised and said only people in Scotland, England and Wales who had registered as donors using their driving licence application form were potentially affected.
Mr Burnham said a system had been put into place to prevent the error from happening again.
Burnham 'regret' over donor error
While many of the 17 million registered donors give consent for all their organs to be used for transplant after their death, some have withheld consent for certain organs - such as their eyes.
For the last 10 years however, the details of some donors' preferences have been wrongly recorded because of a technical error. The blunder only came to light in 2009 when NHS Blood and Transplant started to write to donors to check their organ donor preferences.
Many donors wrote back to say the information was incorrect.
Stephen Banks, from Worcestershire, wrote to the BBC: "I renewed my driving licence in March this year and ticked the options to donate all my organs apart from eyes.
Huddersfield University student Stephen Banks agreed to donate organs when he renewed his driving licence.
"I decided I was happy to donate my liver, lungs, pancreas and kidneys.
But a few days later, on 17 February, I received a letter from the NHS thanking me for donating and telling me I had also donated my eyes.
I was shocked as I knew I hadn't agreed to that.
I didn't know how to correct the mistake and I'm too embarrassed to call them and say 'You can't have my eyes.' So, at the moment I'm donating everything.
I think the register is a good idea as it can give people an extra chance at life but when mistakes like this are made, it can lead to a loss of confidence in the service."
"However I then got a letter from the NHS which said I had donated all my organs including my eyes."
"I feel a bit embarrassed to call up and say, 'I want my eyes back.'"
A spokesman for NHS Blood and Transplant said Mr Banks should not worry about contacting the authority to verify his records and that they would welcome his call.
Mr Burnham said a review led by Professor Sir Gordon Duff, of Sheffield University, would be commissioned to find out how the data was wrongly recorded.
The health secretary said: "We do need to get to the bottom of it.
"It would appear to relate to a technical error going back to 1999 and this was how data was transferred between the DVLA (Driving and Vehicle Licensing Authority) and the blood and transplant service. That has now been corrected," he said.
Donor applications originating from the DVLA were suspended in early March.
In a statement, NHS Blood and Transplant said: "We assure everyone currently on the organ donor register that the affected records will not be used in discussions with their family about organ donation.
"They will only be used once they have been corrected in accordance with the donors' wishes."
The body has already corrected 400,000 flawed records and said steps would be taken to contact all those who were possibly affected to confirm the details held about them.
Anyone who was not contacted could be confident their records were accurate, it said.
"There are a small number of cases, 21 over the past six years, where the person has died and their preferences may not have been correctly recorded," the statement said.
"In each case the family gave permission for the donation to take place but it may not have been in line with the individual's preferences.
"We sincerely apologise for any distress this may have caused."
No-one had been registered as a donor against their wishes, it added.
British Medical Association chairman Dr Hamish Meldrum said: "You want people to come forward and donate their organs in good faith so that people who have serious conditions can have their lives saved.
"It does erode the public confidence in that and that's very concerning."
He said the review was needed in order to prevent patients' lives from being put at risk in the long-term.
Joyce Robins of pressure group Patient Concern said she was angry such "sensitive data" had been so "carelessly handled" and she feared it would dissuade people from joining the register and result in people taking their names off it.
She said the health authority should contact everyone on the register and ask if all information was correct.
It is against the law to take organs from the dead without their prior consent, or that of their family after death.
Shadow health secretary Andrew Lansley said consent should always be given and if that had not happened, a full investigation had to be carried out.
He stressed, though, that it was important for people to continue donating.
Liberal Democrat health spokesman Norman Lamb described the data error as "a shocking revelation" and called for a "full inquiry".
He said: "People have to have absolute confidence that their wishes will be respected."
Have you been contacted by the NHS to confirm your organ donor preferences? Have you ever given consent for a relative's organs to be donated? What is your reaction to the NHS blunder?
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