The PS wasn't the standard for home consoles in 1994, 95, or 96. The SNES was.

And besides, why are you counting PS3?

Never managed to make games as good as the best Nintendo-games though

I would dare to say they do that but Nintendo delivers better accessibility by making their games suitable for young and older gamers.

Which disc format did consoles released after PS3? PS4, X1 and WiiU all used BD. So they defined the standard with PS3 as well =p

PS being the standard is incorrect they simply made gaming mainstream but other companies are the ones who have done the standard features in home consoles today:

NEC: First Internal Memory 
Nintendo: First modern console and controller, Practical Motion Control, Shoulder Buttons, Modern Analogue Stick, Wireless Controller, Rumble, Retro Service (VC)
SEGA: Online, CD Format, Memory Cards, Online Connectivity out the box, Analogue Triggers
MS: Integrated HDD, Modern Online Ecosystem, Universal Integrated Chat, Introduced Digital Distribution

No, it didn‘t. Nintendo never follows Sony. On the other hand Sony followed Microsoft which defined standards for online gaming, console shooters, paying for online gaming and indie games on consoles (Nintendo adapted the two latter standards as well).

For all the talk about "Sony" brand meaning so much in the 90s, there sure wasn't much evidence of that early on.

The Saturn was outselling the Playstation quite easily in Japan for example.

The N64 was destroying the Playstation's sales pace in North America and also had a way stronger launch in Japan.

Money-hatting IPs like Final Fantasy and Tomb Raider is when the Playstation started to accelerate in sales, not their brand.

They would have gotten beat by the N64 plain and simple if Nintendo had compromised with their close partners like Squaresoft and Capcom and used CD-ROM. You could still have had cartridge based games too if that's what Miyamoto wanted. They would not have been able to compete against Nintendo's 1st party IP of the day like Mario 64, GoldenEye, Zelda: OoT, Mario Kart 64, etc. on top of things like Nintendo having exclusive Star Wars games and 3rd party titles like Final Fantasy VII, Resident Evil 2/3 (day and date), Dino Crisis, Chrono Cross, etc. 

Hard Disk format. The storage media used to get the game home from the store isn't really as important as the controller standard that Nintendo has set. My maths tells me that the joycon is about to become the most popular/best selling controller ever.

must......not......get.....myself.....banned

I think people underestimate the huge power Sony had as a company back then. They were "the" electronics company. Always setting out for market standards when it came to formats and other media. Some failed hard like Betamax, but then you have Trinitron or the Walkman. And with that power, they put a huge marketing budget behind it. First: moneyhatting Japanese developers (and in the west to some extent), using old fashioned Tokyo relations (as opposed to Nintendo's Kyoto relations) and driving that with massive marketing worth billions today and utilizing their extensive global distribution network. They were definitely not some kind of underdog. They were more like IBM coming into the personal computer market.