Measurement is not the same as wavefunction collapse. A measurement is an interaction that produces information about a system that can be copied to multiple other systems. Such an interaction does not copy all of the information in the state of a system, only the information instantiated in some set of orthogonal projectors, see
This does not contradict the no-cloning theorem, which prohibits copying the whole content of the information in the state of a system, but does not prohibit copying the specific information cited above.
You ask what the measurement operator for one qubit being measured onto another. If they are being measured in the computation basis $|0\rangle,|1\rangle$ then a perfect measurement is represented by controlled not $U = |00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle 11|+|11\rangle\langle 10|$, which is unitary.
If the measured qubit is sharp in the computation basis and the meter qubit is in $|0\rangle$ then the final state of the measurement qubit is the same as that of the measured qubit.
If the state of the measured qubit is not sharp in the computation basis and the meter qubit is in $|0\rangle$, then we get
The way to interpret this is that there are two versions of each qubit, one in state $|0\rangle$, the other in state $|1\rangle$. As the meter qubit interacts with other systems they too will differentiate into two versions, one for each possible outcome. These different versions of each system don't interact with each other, so you can't see them. Since the resulting versions each act approximately like the universe as described by classical physics they are called universes. This all takes place unitarily. See
The old state is not erased, rather it was initially prepared in such a way that the measurement gives you the information you want.