Interaction between a physical object in a superposition, and another object, usually causes the second object to also go into a superposed state, that will be correlated in some way with the original superposed state. For example, this is true of the quantum-mechanical description of a measurement interaction - some part of the measurement device has to physically interact with the measured object - and the situation where the experimenter opens the catbox falls into this category in a complicated way (the eyes of the experimenter become entangled with the quantum state of the interior of the box, which will include photons that exit the box).
For a good measurement, you want the measuring device to have multiple states which are cleanly correlated with the different quantum states of the measured object. That way, the state of the measuring device can be regarded as a reliable proxy for the state of the measured object - e.g. if the Geiger counter clicked, it should mean, with close to 100% probability, that a charged particle really did pass by.
To put it another way, the whole system of detector+object, if described quantum-mechanically, should go into a superposition which mostly consists of "object in state i, detector shows state i", summed over i, plus a small contribution from mismatch states like "object in state i, detector shows state j". There is a whole branch of quantum mechanics called quantum measurement theory, which analyzes the physics of measurements.
The many-worlds theorists certainly want to regard this as a situation in which there are now multiple worlds, one for each state of the detector. But when you try to argue this in detail, it falls apart somewhat. There isn't an objective way to divide up the total detector+object superposition into the alleged worlds; and when you try to do this, the probabilities come out wrong - some observations should be more common than others, yet you just get one world for each possible measurement, implying that they are equally common.
(In fact, in the most rigorous version of many-worlds theory, "many interacting worlds", all the parallel worlds are actually distinct from the beginning to the end of time - they don't actually split in two. Instead, they converge and diverge in similarity.)
By the way, the original and true version of the Copenhagen interpretation does not say that the cat is both dead and alive until you look. It just says that you don't know, until you look. The quantum superposition is not interpreted as the reality, it's just a statement about possibilities. That might sound simple and sensible and fair, but the problem is that it is very hard to come up with a theory beyond quantum mechanics, that gives the same predictions, and which also makes definite statements about what the physical reality actually is, at every moment.