In general there're a lot of PS franchises although PS don't us a lot of them. Even without mentioning PS1-PS2 IPs there are a lot PS3-PS4 titles which have become frozen. Why they don't return old franchises is a big question. Maybe they don't want to bring back some older stuff and want to feel more modern in terms what they do.
Anyway, even without some of their IPs they have a lot of partnership deals. During PS5 days there were some timed exclusive games from some partners e.g. Deathloop, Ghostwire: Tokyo, Kena: Bridge of Spirits, Godfall, Sifu etc. In 2024 you may remember Rise of the Ronin, FF7: Rebirth, Helldivers 2, and don't forget about upcoming Silent Hill 2. Some of these games which are mentioned were initially released only on PS5 or on both PS5 or PC but they all have played some role in the current console race. Some people have chosen PS5 over XS not only because of God of War: Ragnarok or Gran Turismo 7 but these games also attracted some audience. It's often impossible to rely only on 1st party stuff. TBH Xbox console today has (timed) exclusive games. The problems occur when some of these games become controversial or get mixed reception e.g. Battletoads, Bleeding Edge, Redfall, Crackdown 3, Halo: Infinite. The problems occur when these games release and there isn't another option to play. Some partnership deals is the thing which solves such trouble. When Godfall initially released there was Demon's Souls, when Ghostwire: Tokyo released there was Gran Turismo 7 or when there weren't 1st party games, there were Rise of the Ronin and FF7: Rebirth. Xbox also has some deals with a few stidios like Ori games or Flight Simulator games but sometimes it has less such titles.
Therefore I would say while there are a lot of 1st party franchises which PS don't use, they have more stuff from partners which increases PS library.