I don't know what Strava does behind the scenes but they do seem to try to stick your GPS track onto their segments.

Elevation is the least accurate output of GPS just because of the way the system is designed to emphasize location. From your point on the Earth your receiver can probably "see" nine of the satellites, which are broadcasting their location and their atomic clock time. Your device calculates its position by comparing the light speed delay in each clock signal to find its distance from each one. Most of the satellites your device can see will be nearer the horizon, so they don't have a lot to contribute to the elevation, and the ones that are overhead are closer, making a smaller delay to measure.