Yes, actually, spinning wheels were normal household objects. Spinning yarn was a typically female work done during the winter months, and even noble women were doing it, especially than it was one of the few actions that could be performed by pregnant women. Look at many other fairytales, from Briar Rose to Rumplestiltskin. The English word "spinster" originated as a phrase for someone who has nothing to do but spin the yarn (and the metaphorical use of the phrase, regarding telling tall tales is also... telling, that's what you did when spinning).

If you're looking for symbolism, you can look at this this way: the princess usually does not know what a spinning wheel is - because she's a pampered child. Once she's introduced to labor, her life ends, and must be regained by her man, regained with a kiss (and in many versions of the tale it was sex). The prickling of the finger can symbolise menstruation, or loss of virginity (the spindle is a phallic object, and actually few spindles were sharp). Creepy, yes, but that's fairy tales for you.