The Clearsprings Group, which runs the service, said they would no longer use the red bands after migrants said they had to hide them because they made them feel uncomfortable on the streets.
The bands have been likened to the yellow stars that Jews had to wear in Nazi Germany.
Residents at Lynx House will be given photo identification instead.
Dave Smith, from the Christian refugee charity Boaz Trust, told Premier's News Hour: "I think there needs to be far more training.
"When the government gives out contracts for things like asylum accommodation, it often goes to the lowest bidder.
"It is very insensitive I think, to say the least, because when people arrive in this country, if they've come from a background of persecution and so on, the last thing they want is to be pointed out again as being different.
"They want to be treated normally [and with] dignity."
The controversy comes after asylum seekers in Middlesbrough said they were abused after their doors were painted red.
A company housing the asylum seekers later said it would repaint the doors after concerns people were being singled out as immigrants.
Mr Smith added: "But [the wristband issue] is different from the red doors in Middlesbrough.
"The red doors in Middlesbrough were just horrorfic and totally unnecessary, and quite racist really because it just says, 'Come and throw bricks through out windows. These are asylum seekers living here'."
