To a certain degree, yes. The material, thickness, and hardness of a pick will have an effect on how percussive the “pluck” of a string will be, which will affect your tone.
However, other than that, the specific choice of color, size, and shape has more to do with aesthetics and comfort than anything else. Once you choose a preferred thickness and hardness for your comfort and style, feel free to experiment with different materials and shapes and colors — notice how each affects your sound and how you play. We know guitarists who use several different picks. A thin pick for strummed parts, a small solid pick for leads, another for acoustic guitar, and so on.
Remember, everything involved with playing guitar or bass functions as a system, from the guitar to the amp, the strings to the cables, the pedals to the pick, it’s all part of the tone generation process to one degree or another. Will there be a dramatic, extreme difference between the sound of two picks? In a word, no. But there will likely be a subtle difference.