Nope! Different active (vibrating) lengths are involved. Because the strumming position doesn't change, a shorter (fretted) string is strummed relatively closer to its centre. This affects the number of audible harmonics produced. Try strumming an open string at the 12th fret. It'll produce a relatively dull tone without the second harmonic. Of course the position of the pickup(s) plays an equal role.
Also the length/gauge ratio isn't equal, which makes this even more complicated. A shorter piece of string means less mass, so less energy. That's why I chose equal fret positions (2nd) for producing the sound samples. The only variable was the string gauge, resulting in different tensions.
It's a complicated matter with many variables. Not to mention how the behaviour of the neck and body, with a partly vibrating string compared to an open string, affects the composition of the tone.