That power supply looks like it’s for the laptop, not the monitor. I’ve never seen a monitor with an AC/DC adapter. It’s always a three prong plug that goes directly into the monitor.
If that’s for the monitor, then having only two prongs is common for a DC adapter, and that is not the problem. Weird that your plug has a place for the third prong, though.
Still, even if it’s not the cord, there is something wrong with your monitor. It is not properly grounded and is putting out a shock. That shock is traveling to your laptop. When you plug in your laptop, you ground it, and thus ground the monitor.
So what you need to replace is the monitor. It’s unlikely you’ll be able to repair it.
In theory you could ground the monitor yourself by putting a wire on anything metal and hooking that to ground. But that is nothing more than a temporary solution.
I have a flatscreen monitor that comes with a power adapter rather than a standard power cord. It’s only a few years old so I think they still make them on occasion.
I agree, replace the monitor. It is actively trying to murder you.
If I was the OP I’d also be wondering about the building wiring where I am. It might be informative to test that. you may have a significant wiring difference between the outlet the monitor plugs into and the outlet the laptop plugs into.
If a UPS, power strip, etc., is in the mix, that could be faulty too.
I wonder which country the OP is in? That’ll help us know about typical voltages, likely quality of electricity & wiring, etc.