In this case the thing to think about is how the energy to vibrate the string is transferred to the string. In the case of your question that is plucking, but it is instructive to look at something different.
Another way of causing a string to vibrate is to use air-transmitted resonance.
You can set up a sound generator to emit pure sine wave that is equal to, say, the ground note of a particular violin string. If that sound is sufficiently loud the string will pick up that vibration. If the source is a single frequency the string will resonate just that frequency. If the driving sound is a pure sine then a string will not spontaneously start to vibrate at some harmonic of the driving frequency.
As you would expect: if the driving frequency is equal to some higher harmonic of the string's fundamental frequency (and it's a pure sine) then the string will resonate at that frequency only.
Note that in the case of resonance transferred by air the transfer of energy to the string is distributed over the entire length of the string. It's a very mellow way of transferring energy, so to speak.
Conversely, when you pluck you apply the force at a specific point along the length of the string. Because of that sharpness of location the effect is that you cause vibration of multiple of the harmonics that the string can vibrate in. In effect you drive multiple of the harmonics simultaneously.
The fact that the string has fixed endpoints is relevant only in an indirect way. I mean, in order for the string to be able to vibrate at all it must be under tension, and the only way to be under tension is to be fixed at both ends.