App id vs. Game id for Android Admob and GAme services

On android, I’m a little confused whether to add the game service id (under project settings )from the Play console or to use the app id from adMob as the app id?

You will much more confused when OAUTH2 issues. But let me tell you.

Game service ID is in the link of google play console when you enter application (Check HERE at below.){HERE}

Rest your problems related with authentication, please adb logcat to your device for logging and troubleshoot.

About your admob problems, you dont need admob id anywhere. You just need id of your ad type. They are located under

AdMob -> Apps -> (Your App) -> Ad units.
You will create add units there and put their id into project settings admob array.

So the game service ID is the number you put in the Games App ID field?

Not game service Id, its at the link check {HERE} at above link carefully, its suffix is like appid=127126461264 -> This part what you need.