This is my first time playing a game from these series, so after installing the first Ys I started by running the configuration tool.
One option I noticed was the greyed out "Connect Ys I to Ys II" in the Graphics tab. I figured it may be something like the "World of Xeen" and I tried installing the second part, but the option in the menu didn't activate.

So, my question would be if this option is obsolete/unavailable in GOG version and if it's not, how can it be activated?

I am also interested in knowing if there is a solution to this.

By editing the configuration file manually, the option can be turn on.

But this isn't a big deal. The two software are separated games. In the Eternal Complete, and latter in the Cronicles this option only do what Erpy said:

Completing Ys I will automatically start Ys II without the main menu, and with the same grafical and audio settings like the first game.

Huh, apparently the option works for those who own it on Steam, so Xseed/GOG screwed something up. The two installers should each check to see if the other game is installed and if it is, set up the config file correctly.

I own the Steam release and if I remember correctly, the option was initially grayed out even after installing Ys II. And if I still remember correctly, the option became available after running the configuration tool for Ys II. Has anyone tried this?

Actually I installed both games month or two ago and the option became available without any additional patches after running each of the configuration tools. However, it sounds that the connect option isn't enabled correctly in the GOG.com version.

On the other hand, that probably isn't a big loss since moving from Ys I to Ys II through the connect option apparently doesn't offer any bonuses or other advantages over starting a new game of Ys II from the main menu. And unlike the version for PC Engine/TurboGrafx-16, the stats of the main character aren't carried over from Ys I, either.