Old PC

[Galaxy01]

I have got this old PC: 
MB:  Asrock Z77 Extreme4   with onboard Graphic, and Sapphire RX580 graphic card installed.

I could start up the PC by connecting to the onboard graphic without having to unplug the RX580 card. 

Of course, I could also start up by connecting to the RX580.  In short, I could connect to either of the graphic port.

All along, I have this in mind that I need to unplug the video card (RX580) before I could use the onboard graphic!   Otherwise, the PC won't start up.

Am I outdated or there's some setting in the BIOS that allows to use either onboard or added graphic?

[king1]

usually the onboard graphics isn't functional if an external card is present, this would seem to be an oddity that you can leave the card in and still use onboard...
There should be a bios option for primary graphics adapter or similar for choosing which to use

[Dugimodo]

It's been possible to use both for a very long time by changing BIOS settings, it's just normal for onboard to disable by default when a graphics card is detected as most users only want one or the other. On older systems it was harder to get working but still possible. Sometimes it just depended what you plugged your monitor into as you described. A common game performance problem used to be caused by people accidentally using the onboard after reconnecting a monitor. I used to use onboard + add in for dual monitors to improve gaming performance but honestly it never made much difference. It's also useful so you can use intel quicksync for encoding while using a different GPU for everything else.