I was thinking about the probabilistic nature of quantum mechanics and that if I measured the position of an electron twice in succession, the outcomes would depend on a probability. However, what if I measured the position of an electron in two separate systems that are exactly the same? Would the outcomes be the same or would they be different?
As you said, the result of the measurment depends on probability. Each eigenvalue (i.e. result of a measurment) has a certain probability of coming out when a some characteristic of the system is measured.
Think of this in an easier example. Suppose that you have a pair of dice. Both are exactly the same, so, for both dice, the probability of each number to come out when being throwed is $\frac16$. However, the two dice are independent, so one outcome is not related to the another one. For example, if you throw the first dice, you may get a $4$. Even though the other dice is exactly the same, when you throw it you only have a probability of $\frac16$ of getting a $4$ (and a probability of $\frac56$ of not getting it).
The same happens when measuring observables in quantum physics. It doesn't matter if the two measured systems are the exactly the same. Getting one result in one measure doesn't imply you will get the same when measuring the another: it all relies on probability.
I will address the title, ignoring the content of the question .
If I repeated a quantum measurement, would it be the same?
This is the method of gathering data in particle physics in order to check with as good an accuracy as possible the quantum mechanical predictions for the interactions. For example one sets up a beam of identical particles impinging on an opposite beam of identical particles and measures one by one interaction the four momenta and charges of the new particles produced at each interaction.
This is a repeated quantum measurement, and each event is different. According to the theory to be tested, one chooses classes of events out of the millions, as in the picture, when looking for Higgs to two photon events, as an example. The accumulation of similar events gives a distribution which is compared with the probability distribution from the quantum mechanical model for the reaction proton+proton giving a Higgs meson that decays to two photons.
The individual events differ in number of particles and momenta, so evidently one does not get "the same measurement". The distribution of the accumulation of events is what is repeatable. A different experimental setup with the same input beam and the same two photon channel search, should give the same distribution for the four momenta of the events. That is what is predictable and repeatable.
in fact , that is why the LHC has two very expensive similar but not "same" experiments , ATLAS and CMS, ( each with over 3000 physicists, expensive setups) to ensure that the "sameness" for the distribution exists, independent of the instruments. Here are their independent distributions for the mass for Higgs to gamma gamma from the two experiments: ATLAS, CMS.