Speaking at a Labour Party conference fringe event, Ms Huq also said: "If you hear him on the Today programme, you wouldn't know he's black."
Her remarks were criticised and Labour suspended her from the parliamentary party pending an investigation.
Tory Party chair Jake Berry called her comments "racist" and "disgusting".
Deputy Labour leader Angela Rayner said the remarks were "unacceptable".
Speaking to the BBC's Politics Live programme, she said Ms Huq should apologise, while the party's foreign affairs spokesman David Lammy described the remarks as "unfortunate" adding: "I wouldn't have made them myself."
A Labour spokesperson said: "We condemn the remarks, they are totally inappropriate, and we call on her to withdraw them."
The Ealing Central and Acton MP, who will now sit in Parliament as an independent, was recorded making the comments on Monday evening at a fringe event entitled What's Next for Labour's Agenda on Race.
The audio clip was published by the Guido Fawkes website minutes before Sir Keir Starmer began his speech to the Labour Party Conference in Liverpool.
During a Q&A session, she said: "He's superficially, he's, a black man but again he's got more in common... he went to Eton, he went to a very expensive prep school, all the way through top schools in the country.
"If you hear him on the Today programme you wouldn't know he's black."
Following the publication of the clip, Ms Huq tweeted: "I have today contacted Kwasi Kwarteng to offer my sincere and heartfelt apologies for the comments I made at yesterday's Labour conference fringe meeting.
"My comments were ill-judged and I wholeheartedly apologise to anyone affected."
Mr Kwarteng, who became chancellor earlier this month, was born in east London and has Ghanaian heritage.
Former chancellor and Tory MP Sajid Javid said he was "appalled and saddened" by the clip adding: "Rather than give encouragement to racists and people who seek to divide us, she should know better. Not too late for her to show that she does."
In a letter to Sir Keir, Mr Berry urged the Labour leader to "join me in unequivocally condemning these comments as nothing less than racist".
The Conservative Party chair also labelled her words as "vile" and "disgusting".
Sunder Katwala, who chaired the event at which Ms Huq made the remarks, tweeted: "My comments from the chair that people should be judged on their politics and ideas, not race, and that Kwarteng being a Conservative did not mean he was not Black, were strongly applauded by the fringe audience."