Is it necessary to take a malaria prophylaxis or at least a stand-by vaccination to visit Zanzibar?

Define necessary. No one is going to report you to the authorities if you don't take meds; no one at customs is going to deny you entry if you don't carry them in. But failing to take prophylaxis seems unnecessarily risky, considering the potential consequences.

There has been enormous progress eliminating Plasmodium falciparum from Zanzibar, through the efforts of the Zanzibar Malaria Elimination Program and other organizations; malaria has approached total eradication several times in the last decade and a half. But total eradication has never been achieved, and unfortunately, cases are still reported.

The official guidance from the US Centers for Disease Control and Prevention (CDC) for Tanzania remains

Areas with malaria: All areas <1,800 m (5,906 ft).
Estimated relative risk of malaria for US travelers: Moderate.
Drug resistance4: Chloroquine.
Malaria species: P. falciparum >85%, P. ovale >10%, P. malariae and P. vivax rare.
Recommended chemoprophylaxis: Atovaquone-proguanil, doxycycline, or mefloquine.

with no exception indicated for Zanzibar.

The UK Foreign Office's foreign travel advice notes malaria is common to Tanzania, and the NHS FitforTravel website advises that Malaria precautions are essential in all areas below 1800m, all year round; similar cautions are given by Australia's government. The UK National Travel Health Network and Centre explicitly states

There is a risk of malaria in Zanzibar.

Germany's Ministry of Foreign Affairs (Auswärtiges Amt) also cautions travelers that

Ein hohes Risiko besteht landesweit unter 1800m inklusive der Städte und Nationalparks. Ein geringeres Risiko herrscht in den Höhenlagen zwischen 1800 und 2500m, in Daressalam und auf den Inseln Sansibar und Pemba. In den letzten großen Regenzeiten 2013 und 2014 kam es zu einem deutlichen Anstieg der Malariafälle in Daressalam. Eine Malariaprophylaxe mit Medikamenten (Chemoprophylaxe) ist daher ratsam.

(Google Translate) A high risk country under 1800m, including the cities and national parks. Lower risk exists in the altitudes between 1800 and 2500m, in Dar es Salaam and on the islands of Zanzibar and Pemba. In the last major rainy seasons 2013 and 2014 there was a significant increase in cases of malaria in Dar es Salaam. A malaria prophylaxis with drugs (chemoprophylaxis) is advisable.

The Zanzibar Commission for Tourism appears to reproduce information from the CIA World Factbook, which again states

Malaria is still prevalent in East Africa and so one should also take a malaria prophylaxis. There are many different kinds of medications for Malaria. However, precautions should be taken to avoid being bitten by mosquitos. Apply insect repellent and sleep under mosquito nets at night. Wear long-sleeved shirts and trousers in the evenings.

Given the low cost (by Western standards) of malaria prophylactics, and the fact that your trip is only 10 days long (meaning a shorter regimen, so less cost and fewer side effects than a long-term stay), it seems silly to take on the risk. It is a nasty disease that has proven exceptionally difficult to eradicate; why tempt fate? Avoid mosquitos and take your meds.

Though it's not advised to take prophylaxis for extended periods, you would not want several days of your holiday to be spent sick in bed because you didn't take your prophylaxis.

Sure, taking prophylaxis will not prevent you from getting malaria, but it will lessen the risk.