It's improbable to get a payload into a perfect orbit, but obviously there is room for error, otherwise payloads in GSO would not work as well as they do. If the target eccentricity for a payload in GSO is 0, what is the margin of error? Have there been any egregious errors when maneuvering a payload to GSO?
Once in GSO, the eccentricity is perturbed by a few effects, particularly the pressure of sunlight (not the solar wind), on the satellite. It accelerates the satellite at 6PM local and decelerates the satellite at 6AM local, forcing the orbit to become elliptical. The eccentricity causes the satellite to oscillate east-west on a 24 hour cycle. Its control is part of the normal east-west stationkeeping. Generally the box is $\pm 0.05^\circ$ in longitude, but this can vary. A reasonable target eccentricity is $0.00028$ in this range.
@Stu, here is an example of a satellite in GSO, and the impact of slight eccentricity coupled with high inclination with the JAXA description of QZSS .
With that example in mind, you can almost visualize the space of trajectories with varying eccentricity in GSO. If an egregious error were made, you could expect to see variations in both, but will depend on the magnitude the error creates in the initial conditions of inclination and eccentricity. Other perturbations could make the situation only go downhill (pun?) from here, based upon the type of error made and ability to control north-south / east-west station keeping. An older (1982) UT Austin conference paper on inclined/eccentric GSO is cited here