I have Visual Studio Community 2015 on my laptop, but it seems Microsoft Test Manager (MTM) is not installed while installing it. How can I get it alone without reinstalling Visual Studio?
Microsoft Test Manager isn't part of the free Visual Studio Community edition and has to be paid for. It also isn't part of Visual Studio 2019 anymore. Seems Microsoft decided moving test case management to the cloud, calling it Azure Test Plans, part of Azure DevOps. Detailed information can be found in the links below.
Azure Test Plans
Visual Studio edition comparison
Test Professional subscription
I would recommend to have a look at Azure DevOps, I think you'll enjoy it.