Originally, chess pieces weren’t abstract at all. They quite distinctly resembled the things for which they were named. For example, the king was an obvious king, seated on his throne. Today, however, the standard chess pieces, which are called Staunton chess pieces, are highly abstract, and someone with no knowledge of the game would ever guess what the pieces should actually be called, with the possible exception of the bishop and, if lucky, the knight, although “horse” would be a more likely answer.
The reason for this is as follows. Chess originated in India as shatranj. From there, it reached Europe via the Muslim world. Islam forbade the depiction of human images, however, and so when the game reached the caliphate, the pieces had to be changed from their original form. They were made abstract, losing much of their immediate recognizability in the process. It was in this form that Europeans first discovered chess, and whence they got their ideas of how chess pieces should look.
Certain sets still contained pieces that were more distinct and clearly resembled their namesakes, but by and large, the standard has ever since been abstract pieces, and so it is today.