Earlier quoted context omitted.
After studying the slides, it looks like they do a Diffie Hellman-like exchange of user identifiers. g_i = Google's identifier for customer i. s_j = Merchant identifer for customer j. If g_i == s_j then customer i == j. These identifiers might be phone numbers or email addresses or other identifiers that both parties have. Neither party wants the other to learn any identifiers it doesn't already known about. So Googl…
It's not a new and a very common process. A personal identifying variable is calculated in the same way on both sides (usually by a third party), then hashed. Hashed values are then compared/matched.
Furthermore, although reversing a one-way hash is problematic, computing hashes for known inputs (such as an email address) is straight-forward. So it is possible to just test or probe datasets for known identifiers and persons. The potential for hash collisions is of little comfort.