audio mixer - is ground isolation needed for each channel?
No. In fact, the mixer usually functions as the central grounding tie point for the entire system.

For the inputs in particular, which are mostly connected to microphones that have no other ground reference, this is particularly important.

Only in special cases, such as taking an input from a separately-powered instrument amplifier, would you consider using a transformer or other form of isolation (i.e., a "direct box") to separate the grounds.