* Lip: This clearly refers to the part of the body it's applied to.
* Stick: This likely refers to the original form of the product. Early lipsticks were often solid sticks of colored wax or pigment that were applied directly to the lips.
So, while the name might seem a bit odd when you think about it, it's simply a description of what it is (for the lips) and what form it historically took (a stick). It doesn't imply that your lips are glued together!