Western musical tradition - and education - emphasises the importance of harmony. An individual student therefore needs an instrument on which they, unaccompanied, can play with a high degree of polyphony (in the sense of playing many notes at the same time, not the harmonic technique). The piano, with its popularity in the late 19th and early 20th century, was one of the first relatively accessible and affordable instruments to offer this.
The piano has a number of other useful practical characteristics - upright models are portable enough to easily move around a room or from room to room; The piano is loud enough to fill a concert hall, or accompany any other instrument. It also requires relatively little day-to-day maintenance.
Skills learned navigating the piano keyboard are easily translated to other keyboard instruments such as organs and harpsichords, and more recently, controller keyboards for electronic instruments (well-performing examples of which can be built much more easily than string and wind controllers).
Dom's answer already points out the way the piano keyboard maps well to standard notation and the most commonly-used sytem for naming notes and keys.
It could also be said that some of the areas in which the piano is less capable are also areas in which there is less emphasis in Western musical education. for example, the piano keyboard emphasises the diatonic scale, which can lead to a reluctance to analyse music from a non-diatonic perspective; the piano is only capable of limited timbral control, which corresponds to a lack of emphasis on study of timbre; and the piano is capable of fixed pitches only, which means it is difficult to explore territory outside the 12 tone scale.