ASP.NET Core already has this feature. There is a ILoggerFactory to create a logger. You are able to inject the ILoggerFactory to your component (Controller, Service, etc.) and to create a named logger out of it. During testing you are able to replace this factory with a mock, to not test the logger as well, and to not have an additional dependency to setup.
Recently, we had the same requirement in a classic ASP.NET project, where we used Ninject to enable dependency injection and log4net to log all the stuff we did and all the exceptions. One important requirement was creating a named logger per component.