Learn Turkish: İki tane Türk kahvesi istiyoruz.

This is an indefinite noun compound! In Turkish, when you combine two nouns to create a specific concept like 'Turkish coffee', the first noun stays plain (Türk) and the second noun gets a possessive suffix (kahvesi).

Where does the -si come from at the end of kahvesi?

The root word is kahve (coffee). Since we need the 3rd person possessive suffix -i for our noun compound, and kahve ends in a vowel, we must add the buffer letter -s- between them to make it easier to pronounce.

What does tane mean in this sentence, and is it required?

Tane literally means 'piece' or 'item'. When counting things, Turkish speakers often use it after a number. You could just say İki Türk kahvesi, but adding tane sounds very natural when ordering at a cafe.

Since we want two coffees, why isn't the plural suffix -ler used here?

A strict rule in Turkish is that if you use a number before a noun, the noun must stay in the singular form. Because we already said iki (two), adding a plural suffix to the noun is grammatically incorrect.