Eventually all these stdio functions , either for C or C++ sink down to the _write "system call". So you'll need to implement the _write rather than __io_putchar. The 1st arg of _write is the "file descriptor" number. As usual 0= stdin or cin (not writable, of course), 1=stdout or cout, 2=stderr or cerr & clog. You can define more custom descriptors from 3 up.
Note that ITM itself has multiple sub-channels, 0 is the default "ITM print"; the CMSIS function ITM_Sendchar sends to this channel. In CubeIDE you can open more ITM windows for channels 1,2 etc. and send output of each C/C++ handle to separate ITM channels -> IDE windows. For this you implement replacement of ITM_Sendchar.
Good luck!
