The pair, both 18, suffered burns to their faces, chests and hands after the corrosive substance was thrown in their faces in the late night attack in Stone Town, in the west of the island.
Witnesses said two men threw acid on the Britons as they walked through the capital's narrow streets, before speeding off on a moped.
Ms Trup is from affluent Hampstead and Ms Gee was formally a student at the Francis Holland School in Chelsea, The Evening Standard reported.
The pair were in the final week of their three week trip volunteering with the NGO Art in Tanzania.
Television images showed one girl obviously in pain in the back of a car at the Zanzibar airport.
They were flown to hospital in Dar es Salaam, Tanzania, and have since been discharged and police say the women were lucky to escape serious burns.
They may fly home as early as today for treatment, according to local police.
A spokesman for Art in Tanzania said representatives were today at the Aga Khan Hospital to help the girls, who were also being interviewed by Foreign Office officials and again by police.
Zanzibar Assistant Commissioner of Police Mkadam Khamis told The Evening Standard: 'The doctors said the injuries are relatively minor. There is discoloration but they are not expected to be scarred. It is very fortunate.'
The pair had been volunteering at a local school with the organisation i-to-i Travel.
A spokeswoman said: 'There is a chance that attacks could happen again and we are making sure people are fully aware of the protection procedures.'
The company said in a statement: 'All our efforts remain focused on ensuring they are supported whilst assisting them and their relatives with the arrangements for their return home.
'The motive for the incident is as yet not known and we will await the report from the local authorities in Zanzibar before any comment can be made.
'The female clients had been volunteer teachers on Zanzibar and were in the final week of their trip.'
The company, based in Tunbridge Wells, Kent, runs gap year trips to parts of Africa, Asia, central and South America and Australia, with trips to teach in Zanzibar starting at £669 for two weeks.
'The motive for the attack on the volunteers, aged 18 years, has not been established.'
Mr Khamis Mkadam said: 'The incident occurred when the streets were deserted as most people were breaking their Ramadan fast.'
Zanzibar, a paradise island around 22 miles off the coast of east Africa, and part of the republic of Tanzania, attracts thousands of British tourists a year.
Police on the island say it is the first time a tourist has been attacked in this way.
However, there have been concerns that religious tension in Zanzibar have increased in recent months.
Said Ali Mbarouk, Minister of Information, said: 'We should cooperate with other government sectors to ensure that the perpetrators are arrested and brought to justice.
The Foreign Office travel advice for Tanzania warns that although most visits to the country are trouble-free, 'violent and armed crime is increasing'.
The advice, available on its website, says: 'Mugging, bag snatching (especially from passing cars) and robbery have increased throughout the country.'
It adds: 'In Zanzibar incidents have taken place in Stone Town and on popular tourist beaches.'
The police described the attack as 'an isolated incident', refusing to link it to rising religious tension on the island between majority Muslims and its Christian population.
There have been a series of attacks on Muslim and Christian clergy on the island over the past 12 months.
In November a Muslim cleric was injured in an acid attack and last year four churches were set on fire.
Dr Mike Jennings, a lecturer at the School of Oriental and African Studies in London, said Muslim group called Uamsho, which translates as 'The Awakening', was a political rather than terrorist organisation and had no known links to Al Qaeda.
He said they want an independent Zanzibar and to introduce Islamic law on the island.
Dr Jennings said: 'It is political with a religious tone. Some people see it is as terrorism, but I don't think it is.
'Maybe this group was behind the attack, although it is too early to say.'
The majority of islanders are Muslim, with Christians making up as little as five per cent of the population.
A spokeswoman for the Foreign Office said: 'We are aware of an incident and are providing consular assistance.'
She added that officials became aware of the incident last night.
INCREASING TENSIONS ON THE PARADISE ISLAND OF ZANZIBAR
But it comes amid rising tensions between Muslims and Christians.
In November, Muslim leader, Sheikh Fadhil Suleiman Soraga, was hospitalised with acid burns.
Two Christian leaders were killed early this year in separate attacks.
In February, Catholic priest Father Evaristus Mushi was shot dead by two men on motorcycles on the last day of a festival promoting religious tolerance.
The Bishop of Zanzibar said members of the clergy are terrified of further attacks.
Five churches were also torched last year.
A separatist group in Zanzibar, Uamsho (Awakening), is pushing for the archipelago to exit from its 1964 union with mainland Tanzania, which is ruled as a secular country.
Uamsho wants to introduce Islamic Sharia law in Zanzibar.
Supporters of the group have engaged in running street battles with the police in the past, but authorities have not linked the group with the attacks on Christian clerics.
TUMESHIRIKI NA KUCHAMBUA KUTOKA DM''