It has now been revealed that Fire Emblem Fates retains the petting minigames, quietly tucked away in a different form than the Japanese versions of the games.
Unfortunately, this directly contradicts earlier information from an earlier preview. Earlier, it was reported that the scene cuts to a fade to white, not entering the minigame at all. This preview seems to have been based on an earlier build, while the information today comes from press working on the review versions of the game.
Based on this review build, the minigame is only possible in certain scenarios. To be specific, you can only play the minigames with whoever you marry, and the games only come up in certain scenarios in your home.
In one scenario, your wife or husband is sleeping, and you are prompted to rub their face to wake them up. If you jam the stylus, you can wake them up rudely and they will get angry at you.
In another scenario, your wife and husband has just come out of the bath and a lot of steam is coming out. You blow on the 3DS microphone to move the steam out of the way.
What about the original scenario, where you invite people into your house to flirt with you? No more petting is possible in this scenario, but you can scan their bodies as you talk to them.
Nintendo already shared a statement regarding the changes to the petting scenes. It seems that Nintendo has made further changes since that interview.
Nintendo has made many hard decisions in localizing Fire Emblem Fates to the West. You can read about more of their decisions here and here.
Fire Emblem Fates will be released for the 3DS on February 19, 2016.