Visual Studio 2017 version 15.7 and higher & Visual Studio 2019 changed things again.
Disabling the following checkboxes will allow you to keep the browser open (doesn’t close after stop debugging) and opens another tab (instead of another window):
Tools > Options > Debugging > General
- Disable “Enable JavaScript debugging for ASP.NET (Chrome, Edge and IE)”.
Tools > Options > Projects and Solutions > Web Projects
- (Visual Studio 2017) Disable “Stop debugger when browser window is closed”.
- (Visual Studio 2019) Disable “Stop debugger when browser window is closed, close browser when debugging stops”.