What's the relation between arduino standard library and esp8266 library?
The WiFi shield and an ESP8266 with any of several popular WiFi services firmwares are different pieces of hardware, each of which can provide WiFi services to another device connected to it, such as an Arduino Uno. Their interfaces are different, despite that they could be used to do similar jobs - offloading the WiFi communications from the Arduino - and the two libraries are written to drive each of them, respectively.