For multi-tape Turing machines, can we assume that each tape can be in its own state independently of the other tapes?
A Turing machine has just one state, even if it has multiple tapes. If you give each tape a separate state, it’s easy to forget that the action of each tape head can depend on the global state and on the character under every head, which is crucial, because it is the only way the tapes can communicate with each other. A multi-tape TM is a single computer with multiple I/O devices, not multiple computers each with one I/O device.

xskxzr’s answer shows that one state per tape doesn’t actually change the computational power of Turing machines. But it’s important to note that this requires the actions of all tape heads to be allowed to depend on every tape’s state and every tape’s current character. Thinking about each tape having its own state is likely to make you forget this, resulting in a less powerful machine. Conceptually, it’s usually not a good way to think about Turing machines.