I am implementing the AspNet identity in ASP.NET MVC application. But when I am going through the online materials I am quite confused about the 2 dlls Microsoft.Aspne
Microsoft.AspNet.Identity.Core has nothing to do with ASP.NET Core, it's just a way to tell that it is the main (hence the word "core") library of ASP.NET Identity (pre-ASP.NET Core version). Since you are not using ASP.NET Core, you have to use Microsoft.AspNet.Identity.Core.
Basically, you have two scenarios: