I've been keeping my head down working on various projects and apparently Microsoft has been busy making some big changes and it's confusing the hell out of me. ASP.NET Core first came onto my radar when I installed Visual Studio 2017 last year and went to create a new project and suddenly had choices of .NET Framework, .NET Standard, and .NET Core. So I looked into them a little and saw that the latter two are, in some way, abbreviated versions of the full framework. I read this post by Scott Hanselman ASP.NET 5 is dead - Introducing ASP.NET Core 1.0 and .NET Core 1.0 I also found this, which steered me away from ASP.NET Core: Choose between ASP.NET and ASP.NET Core. My takeaway was "Core is new, you're fine to keep using the full framework." So I created a new ASP.NET MVC site using .NET Framework 4.6.2 and Microsoft.AspNet.Mvc 5.2.3.
Since creating that project, I found what seemed like some bugs in both MVC and Entity Framework only to find that the only mention of them on the web seems to be in Core and that's the only place they're being considered being fixed.
Now, today, I'm trying to look up the documentation on System.Web.Mvc.Controller and System.Web.Mvc.JsonResult and it's gone except under Core documentation. I did finally dig it up here where it says it's no longer being maintained?
I understood this much about Core and said, "ok, why should I care? I'm going to keep doing what I'm doing and check back on that when it's done." But now it seems that 4.6.2 is not even supported anymore. Can someone explain what's going on? Is my NET 4.6.2/MVC 5.2.3 project now obsolete before I'm even finished writing it? If not, why has the documentation been retired and hidden away? I'm worried some of my dependencies might not even be available for or compatible with Core.
This answer will try to focus only on ASP.NET MVC 5, and as little ASP.NET Core as possible.
Product Lifecycle
Microsoft still supports this (so fortunately it is not a "short-lived" release from the other answer), and there is not yet an end-of-life announcement,
Framework/Product Retirement
ASP.NET MVC 4 July 1st, 2019
ASP.NET MVC 5
So if you like, you can keep using it (like VB6 and classic ASP users, who are still being supported by Microsoft).
.NET Framework 4.5.2 and above are still fully supported if you do check Microsoft documentation on product lifecycle.
Maintenance
However, you do need to notice ASP.NET MVC 5 is on maintenance mode, as development resources are almost all on ASP.NET Core right now.
You do get,
If you monitor the relevant NuGet packages, you should notice that even recently Microsoft updates them to remediate security vulnerabilities and so on.
The notice you saw from Microsoft Docs, in fact emphasizes on the very first sentence "We’re no longer updating this content regularly." That makes perfectly sense as ASP.NET MVC 5 is rock solid so you should not expect new materials to be added any more.
You said "Since creating that project, I found what seemed like some bugs in both MVC and Entity Framework only to find that the only mention of them on the web seems to be in Core and that's the only place they're being considered being fixed."
Well, it really depends on what "bugs" you are talking about. Like I said earlier, security related issues are still being patched, but bugs with workarounds or functional limitation are least likely to get fixed. It is an open source project, so if you really want, you can fix the issues on your own, as last resort.
On new features side, Microsoft does backport some features from ASP.NET Core, such as dependency injection, new configuration system, to simplify migration at certain degree. But don't expect much.
Migration
Do consider migrating to ASP.NET Core if you can.
I found this fragment from the introduction to this article in Wikipedia to be a great sum up of the great answer provided here as the selected answer. The same was originated in this Microsoft's blog post: https://devblogs.microsoft.com/dotnet/introducing-net-5/
(I'll link every technology mentioned, where possible, to its own Wikipedia article, for further reading.)
Final release of ASP.NET MVC: 28 November 2018. ASP.NET Core has since been released, which unified ASP.NET, ASP.NET MVC, ASP.NET Web API, and ASP.NET Web Pages (a platform using only Razor pages). MVC 6 was abandoned due to Core and is not expected to be released. Core is currently planned to merge into ".NET 5".