How is an airline call sign assigned?
These designators are assigned, on request, by ICAO. The request is made by the applicant through their National Aviation Autority. 

There are actually two designators: the ICAO three letter code and the ICAO R/T designator.
All assigned designators are published in ICAO document 8585.

Two things are to be considered:


The three letter code must be unique to the aircraft operator 
The R/T designator (callsign) must not cause confusion with other operators flying in the same area,  preferably at most three syllables long and it must be pronounceable in at least one of the following languages; English, French, Spanish or Russian. 


Since there are many operators that have "airline" or "airways" in their name, that word is usually avoided in the R/T designator as it easily causes confusion.

For pilots it is not too difficult, they are typically paying attention specifically to their own R/T callsign. For controllers it is more difficult since they have to associate the three letter code on their screen with R/T designators that or not related in an obvious way. 

For example:


British Airways has three letter code BAW and R/T designator
"SPEEDBIRD". 
Thomas Cook Airlines Ltd has three letter code TCX, and R/T
designator "KESTREL". 
Thomas Cook Airlines Belgium has three letter code TCW, and R/T
designator "THOMAS COOK".
Thomson Airways has three letter code TOM, and R/T designator "TOMSON"




Since the application goes through the NAA, there are slight differences in the procedure from country to country. Some examples are listed below:


Netherlands


AIC

USA 


FAA Advisory Circular AC 120-26K and 
FAA Order JO 7340.2D.
NBAA call-sign application guidance website