In the Many-Worlds Interpretation of the Mach-Zehnder Interferometer experiment, one interprets the universe as splitting into two, with photons traveling in different directions in each universe.
Not so. What happens is that the photon wavefunction after passing through a half-silvered mirror has two peaks, one transmitted, one reflected. When an observer interacts with the photon, the superposition property spreads to the observer who enters a superposition of states, one seeing a transmitted photon, the other seeing a reflected photon. Each part of the observer superposition is orthogonal to the other, cannot interact with it, and hence cannot perceive it. From the observer's point of view, it is as if the universe split into two separate worlds, with each possible outcome happening in one of the worlds.
The 'as if' is important. It's a pop-science way of explaining why we see a classical world where only one thing happens, selected at random, when quantum mechanics (without wavefunction collapse) predicts that both outcomes do, in superposition.
However, how can a single photon traveling in one direction or another have such a significant impact?
Because of what's called the 'butterfly effect'. Chaotic dynamics in fluid flow has a property of sensitive dependence on initial conditions. Any tiny perturbation gets magnified exponentially with time. It is said that the flap of a butterfly's wings in Brazil can set off a tornado in Texas.
It's then simply a matter of imagining how the weather could affect human affairs - maybe Charles Babbage died young in storm when a tree fell on him, the world followed a different path, all the other people thinking about it died in similar weather incidents, and eventually you were the first to survive. (Although whether you would/could have existed in a world without Babbage is another question.)
While the Many-Worlds Interpretation of quantum physics implies the existence of a multiverse, but my query is: what ensures that every physically feasible outcome materializes within one of the parallel universes?
It depends how you interpret "physically feasible". In a trivial sense, every physically feasible outcome must happen because that's what 'physically feasible' means. The collapse-free evolution of the wavefunction is deterministic. There is only one possible outcome. That outcome is what happens. The only thing allowed to happen by the laws of physics is what does happen. (It's sometimes called the Gell-Mann Totalitarian Principle: "Everything not forbidden is compulsory.")
But if instead you mean "conceivable", then we have to consider the possibility that we can conceive of things that are physically impossible. The initial state of the universe is unknown. All the possibilities that diverged into 'other universes' before we started observing are unknown. Our understanding of what is possible is approximate, and almost certainly wrong.
So while it seems very likely that all the usual 'alternative histories' we typically imagine are possible - if it's just a matter of the weather, or which sperm won the race, or some similar 'random' event - physics offers no guarantee.
