Can the car doing the charging with cables be damaged?
Do you mean jump starting another car? Yes it can happen if there is a problem, or if you alternator was already weak. Too much current drain can cause your alternator to fail but it was just about there anyway.
Only if you hook the cables up backwards… Any other “damage” is largely theoretical…
Modern cars are much more sensitive to power surges than cars of yesteryear. Some alternators don’t do well charging a completely dead battery. And you must not cross up the terminals as you hook up and unhook the cables, just an instant hooked up backward can fry expensive stuff.
I view jump starting as risky. First option for me is a rechargeable jump start box which I have and keep charged. Next choice is a spare battery not hooked up to any car. Last resort is car to car with jumper cables. Even then I get the receiver car going as fast as possible. If I can’t get it fired up in 5 min. or less hooked up car to car then I give up and pull the cables.
I simply would not use one car to charge up the other cars battery. Hook em up, give it 30 sec. for the electrical systems to equalize, and fire up the receiving car. If it doesn’t start immediately wait another min. or so and try again. After the 3rd attempt without success stop and disconnect. That’s my process now as I don’t want to fry my car’s electricals. One car not running is bad enough, two messed up puts me in overload.