The title of the chapter VI of the neukirch's ANT is "Global class field theory", and the first few lines are the following:
the author doesn't explain what is $K$ here, but from the previous definitions it seems that $K$ has to be a number field. So, with this restriction we are not considering all global fields but only the finite extensions of $\mathbb Q$. Why doesn't the author give the definition of adele for any global field? Is it an arbitrary choice?
Adeles and idèles can be defined for any global field. See e.g. chapter 2, "Global Fields", of Cassels-Fröhlich's book "Algebraic Number Theory" , Academic Press, 1967