No, it is not normal, and yes I had that myself. Even touching the GND or connecting to a computer caused it.
The basic Arduino boards are not for an industrial environment. At least the official Arduino boards are good quality, the clones are often not.
You might have bought the cheapest clone, which has a lot of noise on its own and works just enough.
The human body has a certain capacitance to hold some charge. If your Nano board is connected to a computer or a circuit, it might already have a certain potential or charge and you touching it will send a disturbance throughout the whole board.
The way to solve this, is to imagine what is going on:
On the Nano board, there is probably a lot of noise. The environment might be so noisy that every piece of wire is an antenna and gets that noise. Your body has a charge. Your circuit might be more noisy than the Nano can handle.
Go to a open field and power an official Arduino board from batteries. It should not happen there.
Don't use cheap low quality clones, you get what you paid for. You paid for trouble, so that's what you got.
Isolate your Arduino board from the circuit with opto-couplers. There are also usb-isolators and isolators for the 5V power.
Put the Arduino board in metal box. Be careful with the grounding of the metal box. Someone had a serious noise problem on this forum a few weeks ago, and attaching a metal box to the frame of his project introduced noise instead of preventing it.
Pulse stretching sketch (not tested, put down in a hurry)
unsigned long previousMillis; // timestamp
bool enable = false; // enable the millis-timer
volatile bool pulse = false; // variable to send information to the loop()
pulse = true;
attachInterrupt(digitalPinToInterrupt(2), blink, FALLING);
unsigned long currentMillis = millis();
digitalWrite( LED_BUILTIN, HIGH);
previousMillis = currentMillis;
enable = true;
pulse = false;
if( currentMillis - previousMillis >= 200) // stretch to 200ms
digitalWrite( LED_BUILTIN, LOW);
enable = false;
[ADDED] I think the sketch works. Tested with Wokwi: