So it turns out that Rachio was linked to my wife's voice, given that it was her phone and her account. I've since signed onto the iPhone with my GMail account, linked the Rachio with myself logged into the iPhone, and now it recognizes my voice. So it appears that the Google Home can only be linked to Rachio via one account, not two.
Also, there still appears to be an issue with linking Rachio to Google Home on the Samsung S7 Edge. For me, I'm good to go having linked via the iPhone but others may still have issues.