How do we interpret the many-worlds interpretation in daily life? Of course one can say it is futile to reason about that. But if one does could perhaps reveal a wrong understanding of the MWI. Or it could clarfy how far one can extend this interpretation.
Sean Carroll states in Why the Many-Worlds Formulation of Quantum Mechanics Is Probably Correct

So nothing stops us from writing down a state of the form
  (spin is up ; apparatus says “up”) + (spin is down ; apparatus says “down”). (2) 
  The plus sign here is crucial. This is not a state representing one alternative or the other, as in the textbook view; it’s a superposition of both possibilities. In this kind of state, the spin of the particle is entangled with the readout of the apparatus. … Once our quantum superposition involves macroscopic systems with many degrees of freedom that become entangled with an even-larger environment, the different terms in that superposition proceed to evolve completely independently of each other. It is as if they have become distinct worlds — because they have.

Because the many-worlds don‘t interact we will never be able to proof their reality experimentally. But not denying them could produce strange thoughts as in the following story:  
He has bought rail tickets with seat reservation for tomorrow. Before that he‘ and he‘‘ measure the spin of a particle. He‘ measures „spin up“ and he‘‘ measures „spin down“. They know they are in different worlds now which don‘t interact with each other but nevertheless share the same memory. In the evening having not a particular preference he‘ choses red- and he‘‘ white wine. The next morning he‘ and he‘‘ enter the train and sit down on their reserved seat. He‘ wonders if he‘‘ thinks about of him‘.  Then his‘ train starts while his‘‘ is delayd.
Please forgive me for writing down such a trivial if not stupid story. What is wrong with it in the context of the many-worlds interpretation.
EDIT
How do we interpret the following scenario which increases the complexity?: After the measurement was performed his sister dials his number. To whom does she talk, to him' or to him''? If she talks to him' then for reason of consistency she will always have contact to him' in the future. If so, is the decision randomly selected? Is something wrong with this scenario?
 A: A good way to interpret the many-worlds interpretation (MWI) in everyday life is exactly the way you describe it in your story about the man on the train. The man is not able to detect the existence of his alternate self. In everyday life, MWI and the Copenhagen interpretation (CI) are both good approximations to a more accurate description in terms of decoherence. If there had been a situation in everyday life where they were not both good approximations, then decoherence would have been figured out circa 1927 rather than circa 1970, because people would have had easy access to observables that would have pointed them in the right direction.
In reality, it is usually very difficult to observe the detailed process of decoherence except for microscopic systems, typically at low temperatures. Only when such special conditions are present can one observe the process of decoherence. The fact that this process takes a certain amount of time is an example of how MWI and CI can be inadequate approximations, since most versions of them involve something happening instantaneously.
This is all assuming certain versions of MWI and CI. Different people do have different things in mind when they talk about MWI and CI. Some people who talk about MWI mean something much simpler and and more spartan. In this form, MWI is not an approximation at all but rather an assertion of the basic axioms of quantum mechanics, such as superposition and unitarity. For people who have this in mind, CI takes the form of MWI plus an additional collapse axiom, so that any experiment that disproves MWI also disproves CI.
