For now, everything is pretty much based on hints and rumors. Yea, including the names.
In an Interview, Sony CEO said "At this time what I can say is, it's necessary to have a next-generation console" which hinted that Sony is definitely working on a new generation of console.
PS4 has been running for more than 5 years now since February 2013 with its last line of console being the PS4 pro (codename Neo) which was released November 2016.
With Sony PlayStation skipping E3 next year it could be a further hint that they want to get done with the PS4 gen line-ups and be born again.
It is only reasonable to assume that the PS5 would come with the backwards-compatible capabilities which made the Xbox one gen stayed ahead in providing solutions for the nostalgic fantasies of their fans.
It's not certain as to when the PS5 would be released but, it is rumored to be expected in the wide range of 2019, 2020 and 2021. Yea, 3 years is a massive ass range. Like i said, rumored. We'll see

Side note: I hope they're working on a next-gen hand held too

Now for the XBox 2 (codename Scarlett), reports and rumors are scarce regarding this console. But, it is only reasonable to assume that XBox would not watch PS4 sail away and leave it behind, although XBox one X is still the strongest console but, it's still in the XBox One gen. Plus, it isn't a secret that the PS4 beats the XBox one in the department of exclusives. So, Xbox would have to do something massive with the next-gen in that aspect.

According to reports from Robert Jones at T3 a YouTuber had a conversation with XBox's Mike Ybarra:

"He [Mike] said we basically went in there to those companies and said: 'what project do you wanna make?' And they are like 'what's our budget?' And he's 'I don't think you understand, that's not our question, our question is what games do you want to make? We are Microsoft and we have the budget, we just want to know what game you want to do, what are your wildest dreams,'"

So it is now clear that Microsoft are planning for a great comeback and dominance since XBox 360.
Although, it a rumored that the XBox 2 would not come with VR gaming (meh, I don't care about that thing anyways and I only know a handful of people who do).
XBox 2 is rumored to come in 2020. None of these speculations are 100% concrete so it is unsure which would blow which by coming out earlier.

Our duty, keep our eyes open and our ears to the ground or the web or wherever your ears will fit and wait for an amazing experience from the 2 console giants as time goes on.

Thoughts?