Winpty and Git Bash

winpty is A Windows software package providing an interface similar to a Unix pty-master for communicating with Windows console programs.

That is why you need it as described here:

The software works by starting the winpty-agent.exe process with a new, hidden console window, which bridges between the console API and terminal input/output escape codes. It polls the hidden console's screen buffer for changes and generates a corresponding stream of output.

rprichard/winpty mentions:

The package consists of a library (libwinpty) and a tool for Cygwin and MSYS for running Windows console programs in a Cygwin/MSYS pty.

As detailed in "mintty/mintty Tips":

When interacting with programs that use a native Windows API for command-line user interaction (“console mode”), a number of undesirable effects are observed; this is the pty incompatibility problem and the character encoding incompatibility problem.
This basically affects all programs not compiled in a cygwin or msys environment (and note that MinGW is not msys in this context).

As a workaround, you can use winpty as a wrapper to invoke the Windows program.

So:

Why do many tools work in plain CygWin but some of them also need winpty?

This was asked (specifically for Python) in msys2/MINGW-packages issue 2645

there's a non-zero cost associated with using winpty, and most of the time, mintty works fine anyway.
Your problem stems from trying to use mingw python which is a native windows build, and hence expects it to be used from a windows console.
mintty uses pipes for input/output, so it doesn't look like a proper terminal to most programs.

On the other hand, if you use msys2 (or cygwin) python (.e.g /usr/local/bin/python2), everything just works.

(With msys2/MINGW-packages PR 2675, python3 detects the terminal correctly)