Can a transponder be made to give a false ident?........
A transponder doesn't broadcast a fixed code. The pilot (or co-pilot) sets the transponder code as assigned by Air Traffic Control (ATC). The controller would say, for example, "Boeing N12345, squawk four-tree-seven-zero". The pilot or co-pilot would dial the transponder to 4370.
As for giving a "false" identification, anyone could dial a different code but ATC may call you on it. One way to give a false ident would be to have two planes that are flying a similar pattern, especially aligned along the same radian from ATC radar, "trade" transponder codes. Pilot A squawks Aircraft B's assigned code and vice versa. Then diverge and head to each other's destination. Anyone trying to follow Pilot A might be surprised when they realize they're approaching a MiG instead of the corporate jet they thought they were intercepting.
A simpler approach would be to file a flight plan as a different aircraft than you're actually flying, take off from a private runway, (no control tower) pretending you're someone else, and get routed to your destination as though you were a charter flight or something routine versus an escaping VIP.
If you're remote enough, you're not assigned a code until you're under positive control from ATC. If flying under Visual Flight Rules (VFR) in general airspace in Europe for example, you'll squawk 7000. If you don't ascend above 18,000 feet, you could pass yourself off as a general aviation aircraft flying under VFR rules until you've gotten away to a safe distance.
IDENT is a special transponder function that only gets used at ATC's request. ATC will ask for it when they're having trouble picking your 'blip' out of all the traffic on their radar screen, or if you're VFR (which means you're squawking 1200, if you even bothered to turn the transponder on), not on a flight plan, and they need you to squawk a specific code for them so they can keep track of you more easily. This might happen if you're requesting to enter controlled airspace at your destination airfield, or you're picking up a flight plan (can be instrument or visual) after you're already airborne.
When ATC radar sweeps the airplane, its transponder normally replies with that 4-digit number in a 12-bit code (called a secondary return). When you press the IDENT button, it changes to a 13-bit code, and receipt of the extra bit makes your 'blip' on ATC's radar screen pulse momentarily (computer geeks would call this "throbbing").
So all IDENT does is make your blip stand out from all the others that at that moment are on ATC's radar screen.
Can you use a false transponder code? Sure, most transponders have four thumbwheels on the control face that the crew manually sets the squawk code in. They can set it to anything they want from 0000 to 9999. Which might be conflated with a false identity but to IDENT is something altogether different.