Pakish refers to the people and culture of Pakistan. It's suggested use is for Germanic languages. However the word Pakistani is used in local/indigenous languages in Pakistan. Its use is similar to the use of Turkish or Spanish.
The term "Pakish" was conceived and designed as a strategic tool by a Pakish design-thinker Danish Rahi in 1980's and its one of the first official use can be found in one of his interviews with the magazine "The Saturday Post". The term made headlines when it was launched as the title of the book "The Pakish Identity" published in the U.S. and became part of the U.S. Library of Congress archives. Rahi describes the term (Pakish) and discuss its conception in detail in the 3rd part of the book which is compiled of over 360 pages.
The term "Pakish" was designed in collaboration with P.A.W.E.R.R, a think-tank focused on intelligent design and strategic thinking with members in the U.S., Turkey and Pakistan. The term was designed and then officially launched as a nation-branding strategy and P.A.W.E.R.R is also involved in a legal and official use of the term in the private and public sectors of Pakistan. It is also important to note that Danish Rahi never rejects of objects to the term "Pakistani" (more commonly in use), but suggest its use only in local languages (e.g: Urdu) and suggest "Pakish" for use in English / Germanic group of languages.
For more details on this term (Pakish), please read the book "The Pakish Identity"