Stepper motors typically only has 200 steps per 360 degrees. Most servos (even cheap analog hobby servos) have more precision than that.
Yes, servos can be made, and are made, to support 360 degree rotation. Note that "servo" doesn't mean "RC hobby servo" -- a "servo" is any system that has a motor, a position encoder, a motor driver, and a closed-loop controller that combines input commands and the position sensor to determine how to drive the motor.
Here's a servo that has high precision, and does 360 degree rotation (being able to read the position across all 360 degrees): http://support.robotis.com/en/product/dynamixel/rx_series/mx-28.htm
This servo is used in the Darwin-Op humanoid walking robot.