The web page for the Nano is simply out of date and does not reflect the current state of the Arduino libraries which support SPI on this processor and hence board.
In terms of components, it is very nearly just a form-factor shrink of the discontinued Duemilanove, and in the current version with an ATmega328p has capabilities essentially comparable to an Uno.
If we examine archive.org's records for the Duemilanvoe product page, on September 15th, 2010 it said:
SPI: 10 (SS), 11 (MOSI), 12 (MISO), 13 (SCK). These pins support SPI communication, which, although provided by the underlying hardware, is not currently included in the Arduino language.
While on September 26th, 2010 it said:
SPI: 10 (SS), 11 (MOSI), 12 (MISO), 13 (SCK). These pins support SPI communication using the SPI library.
They seem to have simply forgotten to update the corresponding Nano page for nearly six years.