Yakuza was one of the games kat played on the chaos realm channel so I've always wondered if phil's obsession with those games comes from that. Maybe when phil first started talking to her he watched her yakuza videos and they bonded over talking about those games.
Other than that I really don't get it. The yakuza games are long and boring, he falls asleep during them and they get low views and money, but he autistically plays them all to the very end. Any other series would have been ragequit after 1 stream if they received the same results as the yakuza games, but he doesn't ever threaten to quit them.
Also when he brought the fat verison of kat back to the streams, it was to play a yakuza game.
Maybe this is all a coincidence but I don't really believe in coincidences so I suspect phil and kat have some autisitc, sentimental attachment to the yakuza games and it's one of the very few common interests they share.