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Yes the sensor should be replaced, it's part of the safety features on your vehicle.
The steering angle sensor is a clock spring sensor installed behind the steering wheel.
It feeds vital information in relation to vehicle angle to the ESP control module, so in the event of a skid, it will tell SRS, ESP, and ABS control modules which wheel to apply the brakes.
To synchronize means to let the other modules on the car know that the sensor is knew and present.
I don't know that the sensor is bad, the dealer diagnosed it to be a bad sensor, your question did not address that you wanted to know what the sensor is, does, what sync means and if it something you can ignore. And I answered all of that.
let me know if you need any more help. Thanks