PS4, SNES, and 360

I looked it up for myself just now and here's the actual data: between hard-copy releases and digital downloads combined, there have, as of this date, been...

2,180 games released for PlayStation 4
2,103 games released for Xbox 360

You can't just look at the official search results that come up on the official stores because those include a metric done of duplicates due to the fact that those list bundles as separate games. Once you focus on actual unique titles, these are the numbers.

Anyway, the numbers are close! However, the PS4 already has slightly more games at the end of the day, and, unlike the 360, still isn't done. 

However, this said, looking up the total combined number of games that were ever released for PS1, PS2, and PS3, it also looks like your original claim that the early version of the PS3 that had backward compatibility was the system capable of playing the most total games may is probably still factually correct as of yet. The PS4 probably won't catch up to the combined total of all those games.

Also, counting both Virtual Console (659) and WiiWare (463) releases for the Wii, it looks like that system also had more total games than the PS4 so far (a little over 2,600 grand total), although the PS4 may very well yet catch up.

Those are the only two that have more total games though.

Nintendo: SNES, followed by Switch - Good games I've liked from the past, and the other, a handheld that's been alright for me so far.
Sony: PS1, followed by PS2 - Introduced me to survival horror and other classics, as well as providing me with a DVD player.
Microsoft: Xbox 360 - Introducing me to indie titles and shoot em' ups, followed by third person shooters.

Microsoft: Xbox 360
Tbh it's the only Xbox I've played on but it was outstanding and I had an absolute blast on it. Easily has one of my favourite controllers of all time, and a ton of games. Microsoft really matured during this era and became one of the big 3.

Sony: PS2
Best Sony console hands down. The amount of exclusives including third party exclusives were outstanding. Easily the console of choice for that generation. Where as the PS3 and PS4 are superb in their own right, the PS2 was just more distinctive at the time and just an unbelievable library.

Nintendo: N64
Pure nostalgia for me. Banjo, DK64, Goldenye and OoT are among my favourite games of all time. What I love about the N64 is how memorable the games were in a generation where games have aged so badly. Also the console design... slick. The Wii, Gamecube and SNES are really close but I starting gaming on the N64 so it will always be special. And yes I love the controller, FIGHT ME.
I have to say though, the Switch might end up to be my favourite by the time this gen is done, combining everything I love about Nintendo's history but it still needs time before I admit that.

Well that 360 number doesn't include the 3398+ small indie games that where available on the 360 at one point or another.  The total number of games available on the 360 during its entire life span was over 5000+. 

The X1 currently has 2107 games available:

https://en.wikipedia.org/wiki/List_of_Xbox_One_games_(A-L)#F

and that list doesn't include the 575 titles that where added via backwards compatibility so if you add those the X1 actually has more games available then the PS4.

Yeah, that's a good point on backward compatibility. To tell you the truth, I don't often even think about it anymore because I really tend to like the modern smaller games better than the actual classical ones to tell you truth. I definitely own more games on PS4 than I do on any other system personally.

Nintendo: SNES easily the best software lineup during the 16 bit era. Sports game played better on the Genesis/Mega Drive due to having a faster processors that being said one of my favorite sports games of that era NBA Live 95 actually plays better on the SNES. The SNES controller might still be the best Nintendo controller ever made also and one of the best controllers ever.
Sony: PS2 best software lineup of the sixth generation. Plus the DualShock 2 was a solid controller.
Micorsoft: X1 great software lineup which got a nice boost via backwards compatible titles. Game Pass is also a solid addition. The controller is a improvement of the 360 controller especially the the rumble triggers which are great for racing games.

Well I own and played more 360 games then X1 games 205 versus 169 but I have more X1 achievements then 360 achievements and are well on my way to playing more X1 games then 360 games also.

Sony: Either PS2 or PS4. Can't make up my mind just yet.
Microsoft: Uhhh....
Nintendo: GameCube. Even though the Nintendo 64 was the first video game system I ever played on, it was the GameCube that made me a gamer.