Difficult to say for certain; perhaps the footage is old recycled USN footage. But looking at a map of the Strait of Hormuz, it might be possible.

The Strait is divided more-or-less along the centerline between Iran and Oman. The minimum width of the Strait is 30 miles, but not all of the water is useable for large ships.

There are 2 shipping channels on the Omani side, and effectively all large ships must use them. The nearer of the two is perhaps 15-20 miles from Iranian land, but it is just on the border between Iranian and Omani waters.

If the USN decides to transit the Strait with no attempt at secrecy, then the Iranians would be able to track a carrier with little difficulty. Since the shipping lanes are just outside of Iranian waters, flying an aircraft of any sort 10 miles out, while still in Iranian airspace, and waiting is a simple matter. Once the ship of interest passes, simply flying near, and maybe even slightly over the boundary would allow shooting the footage the Iranians say they shot.