A few guesses for you...
On cold start, the GT is going to raise it's idle speed to somewhere in the 900-1200 rpm range. I've noticed too that GT's can have an inherent vibration at these higher rpm's coupled with high (cold) oil pressure. Most likely, this is what you may be describing.
GT's may also have a slightly louder than normal clutch rollover noise... a noise which is common when the clutch pedal is all the way in but which will disappear when the clutch pedal is released.
Finally, you might check the power steering fluid level. GT's have a habit of weeping fluid out of the power steering return hose. If the level gets too low, you could have a noise or vibration coming from the power steering pump which will lessen as the fluid warms up.
Today I went to my workshop and first checked the oil level in the power steering reservoir of my Ford GT. There was oil in it, but it was just below the lower level when it was cold, so I added about 100cc of oil. The oil level was about halfway up the oil level gauge, so I started the engine in that state. The noise stopped. I am very happy. I am truly grateful for your kind advice.
I spent the whole day yesterday searching the internet for information on strange noises in Ford 5.4-liter engines. I found many cases of strange noises coming from the chain due to a malfunction of the chain tensioner in a 5.4L engine, although it was not a Ford GT. Since it was not a metallic sound coming from the chain, I thought it was not the chain tensioner. The Ford GT engine is a dry sump, so if the engine is not started for a while, the oil pressure will drop, and for a few tens of seconds immediately after starting the engine, the oil pressure will be insufficient and the strange noise may come from somewhere. In that case, if the engine is started repeatedly with low oil pressure, the cam and crank metal will be damaged, so I was very worried.
Now that I understand why the noise was caused by a lack of power steering oil, I can calm down and think about it. The Ford GT has an engine in the rear, and the oil line to the power steering rack goes back and forth through the car, so when you start the engine, there is a time lag until the power steering oil returns to the reservoir tank, causing the oil level in the reservoir tank to drop and making it easier for the pump to get air in. This is just my guess, but since the reservoir tank is probably a repurposed tank from another car model, the mid-engine Ford GT's tank capacity is probably too small compared to the return time lag and the rate of oil expansion due to oil temperature.
Currently, I have refilled the oil to near the upper level when the engine is cold, so I will warm up the engine sufficiently as written in the service manual to see if the power steering oil will expand and reach near the upper level. Thank you again for letting me know that the strange noise is caused by a lack of power steering oil. I am truly grateful.
I took a video of the engine starting up, but it turned out to be a pretty ordinary engine starting video that I don't think anyone would be interested in 
