Your mother is cool.

LOL

Joking aside, gaming has been "cool" ever since the arcade was a thing.

Heck, Pong machines were cool for their time.

Sega and arcade gaming was the "niche" cool gaming. When Sony came gaming became cool for adults and teenagers. It was no longer a toy for children.

Sony made gaming more casual. That is all. Their games have always been for those that are less skilled at gaming. This allowed people who couldnt game before to beat games instead of watching others beat them.
Gaming before Sony wasnt "for kids" thats a really simple minded and honestly childish way of looking at it. I remember seeing grandmothers playing Space Invaders and 40 year old business men playing Pacman. Hardly kids.
I was an adult who owned a PS1, but I also owned an N64 too. It was the ease of games and the more casual appeal that Sony brought to the market that had me turn to PC gaming because there was no challenge there. Sure the games are great but they require little to no skill to beat. I remember people telling me that certain games like MGS were just so hard to beat. I played it and while I loved it, I didnt see the challenge they did because I was used to harder games at the time. Still to this day I have PS friends who are envious of how easy I make their hard games look. For me gaming isnt just about bragging rights, beating a game these days is meaningless because anyone can do it. Its the challenging games that make me work for it that I enjoy playing the most

Pah. You pretty much said Sony made gaming "mainstream". Nothing mainstream could ever be cool.

[/sarcasm] 

You would be surprised. Sega would've been the one to do it if Nintendo didn't hold back third parties from them. At some point it no longer mattered for Sega, their fate was sealed. Especially when, "Sega does what Nintendon't".

tagging.

Gaming was put "mainstream" in different times actually depending on the region.
In America games were made popular by Atari and Pong. Nintendo just revived the console market
In Japan games became popular when the Famicom/NES was made popular
In Europe Sony was the first company that was able to sell and sell big in the console market. This was mostly die to Globalization and the fact that sony already had inroads there with their other products.

Another myth is that gaming has gotten more popular. It actually has been stagnant since the NES era if we include the higher income and population growth. In the U.S anyway
Fun fact about 40% of the NES audience was over 18
Check this video at 3:20 if you don't believe me.

Heavy marketing in spaces that were not suited for kids, and 3D graphics. PS1 was to be known as the 3D machine and when developers jumped on board PS1 got more popular.