It does make sense (but it's forbidden, because it is written "on PF request")
And can the PF set his own FCU at your airline ?
Very well. Do this- in the briefing, say "assume any instructions given by ATC to be my request". That should keep you within the SOP and eliminate unnecessary chatter.
Does this little rule ("NO" in the specific page of the FCTM) apply to all airlines or do some airlines have a modified version where pilots have a tiny bit more of freedom ?
Or, do some pilots routinely make this very little violation to touch a knob when they normally couldn't (not ordered to do so)
We seem to have a bit more freedom. The first page of the SOP allows us to deviate from normal operations as necessary to maintain a safe operation. It also specifically says that the PF should make FCU inputs when the PM is busy. I recall on day 1 of my very first airline class, the instructor threw the FCOM across the room to make the point that even though we needed SOPs, the book couldn't fly.
Little violations? Probably. My last flight, I was PF. In the climb with the AP off, as the captain made a PA, we were cleared to our requested cruise altitude. I read it back, and set the FCU. Later on, while the captain was having his dinner, I responded to a radio call, tuned the new frequency and contacted the next sector, with nary a word from the captain. Would my disregard for SOP have caused me to fail a line check at your company?