Its should open the link using your native browser that you set as default.
If you have trouble with that, I advice you to check Epic account preference and create a usable and secure to log into using Epic Games account directly if the problem persist.
Alternatively, you can install Heroic Launcher to manage all your Epic games on Linux. More reliable, less resource consumption than Epic.
I’ve set my default browser to google-chrome.desktop and it works with xdg-open
but nothing happens when trying to login with google in epic games store.
Not sure what you mean with create a usable and secure  to log into?
It should look like the picture below; if this is not happening, I would recommend you to share your system information (get it from lutris Preferences >Hardware Information just the System part).
If you got to the main page of the Lutris website you can look for the Epic Store in the games list, you can also look for it in the Lutris Add Game window. This will show you that there is a script to install the Epic Store with the correct adjustments.
Sooooooo… I don’t have a technical solution, I was able to install and somewhat launch the Brave Browser but it didn’t really work. Then I went to my Epic Store account and “changed the password”, even though I have registered with my google account.
That worked, you can use the google email and the new password that you set to login.
oh, it seems you already had set a password.
When I did it, it asked me only for the new password, now it is asking for the current password.
You could probably reset the password.
Same problem. I click update game list near Epic Icon, box with login method appear, but when I select google, blank window appear and nothing. After closing this blank/empty window, Lutris crash.
My System:
OpenSUSE Tumbleweed, Plasma5, Lutris installed both from OpenSUSE repo and flatpak, AMD APU, NVidia GPU.