It just happened to me that when I turned the engine on, I started to hear a beeping sound with a frequency of about once every 5-10 seconds.

It was not one of the Ford chimes, just a generic beeping sound and I think the source was from somewhere within the dashboard. It was not a loud beeping, if I had the radio on, or had background traffic noise I don't think I would have heard it.

First I checked for warning lights, but there was nothing. I checked whether it was coming from my phone, but it wasn't that either. I turned the engine off and it kept beeping. Opened close the door, and still the same. Then as I was about to take my phone and record it it stopped 🤦‍♂️

Any idea what could have been?

P.S. The car has no aftermarket accessories.

Check your headlight switch - rotate it around and back to auto. This is a known bug with the switch assembly, that finally has a fix after years with this issue. You can get a new switch but the error is very random. I have not had it for months, and then some months I get it more than once.

Right. This makes sense because I think the timing also fits in the sense that it stopped at the same time that the lights automatically turned off.

I guess I can live with it. I have the car 1.5 years and it's the first time that it happens.