It’s not a full SmartThings integration, though. Of course, neither is Alexa. But as of today’s announcement:
Alexa/SmartThings:
Lights: on/off/dim. (No color)
Switches: on/off
Thermostat: up/down/set level .(no heat to cool change)
Routines (Limited to devices recognized by Alexa)
Custom phrases with IFTTT
Google Home/SmartThings:
Lights: on/off/dim/color control?
Switches: on/off
Thermostat: up/down/set level .(no heat to cool change)
Custom phrases with IFTTT
So no native control of locks or garage doors with either.
It looks like virtual switches should work with either, which then gives you a lot of additional options because of the versatility of SmartThings. 
Google Home lets you set the color of Phillips bulbs. It’s not yet clear whether you can set color through the SmartThings integration, though.
Alexa gives you some direct control of some routines, and way more device integration outside of SmartThings.
So both Echo and Google Home display the SmartThings logo. But it’s not a full integration with either.