When bassists talk about tone, the conversations usually center on equipment: what instrument, type of strings, amplifier, and such. Though these things do have an effect on the type of sound created, all these things do is colour the original sound the player creates (as Jaco said: “The sound is in my hands.”)
A great way to explore the myriad of tones you can create is by experimenting with your right hand technique. Play closer to the bridge; notice the brighter sound this produces. Then pluck closer to the neck and hear the thicker tone you get. Do you prefer one sound to the other? Maybe you’ll use them both in different situations, choices that will contribute to your uniqueness. For instance, you may use the bridge sound for the verse of a song and the neck sound for the chorus.
Also, how are you striking the string? If you play fingerstyle, do you pluck with the fingertip? If not, try it. If so, try playing with the side of the finger. Or maybe with the fingernail (I do this a lot to emulate a plectrum sound). If you use a plectrum, do you hit the string straight-on, or at an angle? Are you using just the tip of the plectrum or are you rubbing a lot of surface area against the string? Do you tend to hit the strings hard or do you play with a lighter touch? Whichever you normally do, experiment with the other options and listen for how it affects your sound.
You can also examine your left hand technique. Do you dig into the string when you fret, or use a light touch? Are you playing more with the fingertip or pad? Is your vibrato the classical style of pushing and pulling the string along its length, or the bluesy bending the string back and forth type, or do you use vibrato at all?
The choices you make will determine your sound, and since it’s highly unlikely that another bassist will make all the same choices as you, this will create a tone that others will instantly recognize as you.