Learn Icelandic: Við köllum hann vin.

This sentence uses a double accusative. Verbs of naming, calling, and electing, like kalla, take two accusative objects. The first is the person or thing being named (hann), and the second is the name or title they are being given (vin).

Isn't the word for friend vinur? Why is it just vin in this sentence?

vinur is the dictionary form (nominative case). Because it acts as the second object in our double accusative structure, it must take the accusative form. For this strong masculine noun, that means dropping the -ur ending to become vin.

If we changed this to the passive voice (He is called a friend), would vin stay in the accusative?

No. When a double accusative sentence is changed to the passive voice, both of the accusative nouns shift to the nominative case. The passive version would be Hann er kallaður vinur.

Can I use kalla like this to mean calling someone on the phone?

No, kalla strictly means to call someone by a name/title, or to physically shout out to someone. If you want to say you are calling him on the telephone, you must use the verb hringja í followed by the accusative: Við hringjum í hann.