Reading other comments, a few people have mentioned that the Sony name was huge in electronics back then. This is true but, in the world of gaming, Sony was just a mid tier third party.
As I said before, a lot of big companies tried to jump into gaming but gamers were mostly loyal to the two companies that (appeared to have) dealt exclusively with gaming. Nintendo and Sega. Sega was busy committing suicide by not supporting the Genesis or its expensive add ons.
Early Sony games like Wild Arms and Twisted Metal did okay (EGM even gave it game of the year over Chrono Trigger) but the PS1 was basically sold on its 3rd party library. The biggest games that raised eyebrows were Ridge Racer, Battle Arena Toshinden, Wipeout, Tekken, Mortal Kombat 3, Street Fighter Alpha, etc. and there was no reason to keep any of them exclusive.
PS1 won because it had tons and tons of games that the N64 didn't, it had the best versions of games that the two shared, it had cheaper games (It was also super defective).
Launching late really hurt Nintendo but launching with a more expensive format that guaranteed worse games just handed the industry to Sony and they've run with it ever since. By 1998, it wasn't even a battle anymore.