Learn Russian: Вчера мы раздали каждому клиенту по два купона. - Yesterday we handed out two coupons to each client.

This is the core of distributive quantification for numbers. While по usually takes the Dative case, numbers 2, 3, and 4 are special: after distributive по, they take the Accusative case. Since купон is an inanimate noun, the Accusative of два is just два.

If по puts the number in the Accusative case, why is the noun купона in the Genitive case?

The preposition по only affects the case of the number itself. Once the number два is established as being in the Accusative case, the standard number rules take over: the inanimate Accusative два forces the following noun into the Genitive singular (купона).

How would the phrase change if we gave out five coupons instead of two?

For the number 1, and numbers 5 and above, distributive по goes back to taking the Dative case. You would say по пяти купонов (where пяти is Dative, and купонов remains Genitive plural). Note that in casual speech, native speakers sometimes use the Accusative for 5+ (по пять купонов), but strict grammar requires the Dative.

Does the prefix раз- in раздали have a specific meaning here?