It is best to first understand the uncertainty relation as a mathematical property of the wavefunction of a particle. So what it is saying is that is mathematically impossible to construct a wavefunction which does not satisfy the uncertainty relation.
Then we can move on to think about what physical meaning this relationship may have. To avoid getting bogged down with interpretational questions when first learning QM it is best to stick to experimental (operational) procedures where we can test the uncertainty principle. (This is just deferring deeper questions - not saying that these deeper questions are of no interest or meaning).
Now to relate the theoretical predictions of QM (which have a probabilistic nature) to experimental results we inevitably will need to do multiple experiments with many particles all prepared in the same way so they have the same initial wavefunction. If we do a large set of measurements of position we can calculate a standard deviation of position; if we do another large set of measurements of momentum (on a different set of particles but prepared the same way) we can calculate a standard deviation of momentum. These standard deviations will necessarily satisfy the uncertainty relation whatever method we used for preparing the particles.
I'm not saying that is the end of the story - just a good way to start thinking about it.