So, you can ask your spouse to pick up milk on their way home from work, for example, or remind a housemate it's their turn to clean the dishes. They'll see the request on their phone as a reminder, or on the main screen of the Google Home Hub, as pictured above.
For instance, you can say, "OK Google, remind Todd to pick up cheese on his way home," or even something more specific that includes the location: "OK Google, remind Todd to pick up cheese when he gets to Trader Joe's." Then, Todd will get that reminder when he arrives at Trader Joe's, or when he gets in the car on the way home. You can also add a specific time.
It only works with people in your family group, which you can configure on Google's Family website. Anybody you try to add must accept your invitation — so you can't use this feature to assign tasks to contacts without their permission.
The new assigned reminders feature is coming to phones, smart displays and speakers with Google Assistant installed, starting in the U.S., U.K, and Australia. It only supports English right now.
I noticed it's not yet active for me, on my Android phone or Google Home Hub, but Google said it's rolling out over the next few weeks, so check back if you don't see it yet.