At the present time, echo can run a SmartThings routine. Google home cannot. (You can set up a virtual switch to trigger a routine, but there’s no direct way to run a routine.)
The “OK Google, good morning” is triggering a built-in function of Google Home which has nothing to do with SmartThings. It’s called the daily briefing.
Smartthings will allow you to create a SmartThings routine called “good morning” but Google Home doesn’t know that it exists. That’s why saying good morning to google home is not doing any of the things in your good morning smartthings routine.
You can change a few of the things that Google Home will say by using the “my day” settings. But again, nothing to do with SmartThings.
You could ask in the Google home thread, but I don’t think there is a way at the present to combine the daily briefing with smart home actions.
As far as setting up a virtual switch so that you can run your routine from Google home, see the following. It’s the way we used to have to do it with Alexa. It’s the same process for Google home.