The app, which appears to be unfortunately geo-restricted to the United States, is like an automated version of an article that circulated recently showing folks standing in front of portraits at museums. In many cases, the old-timey people in the paintings resemble them uncannily, but, other than in rare cases, that's not the case at all with Google's app.
Google matched me with someone who doesn't look like me in the slightest, a certain Sir Peter Francois Bourgeois, based on a painting hanging in Dulwich Picture Gallery. Taking a buzz around the internet, other folks were satisfied with their matches, some took them as a personal insult, and many were just plain baffled, in that order, as presented below. From all that, it's pretty clear that deep learning systems like those from Google are great at matching individual details, but painfully miss the big picture.
Update: As some folks, like Twitter user @AndrewCBartlett have pointed out, and as the Google Arts & Culture app warns, the museum-matching feature appears to be limited to the United States. I was unknowingly able able to access it in France thanks to my always-on VPN, so that's another way to go. If it does work in any other nations, please do let us know in the comments.