... but gdb supporting Python, doesn't mean Python on its own can access gdb functionality (apparently, the gdb has its own built-in separate Python interpreter).

Is there a GDB for Python?

A set of GDB macros are distributed with Python that aid in debugging the Python process. You can install them by adding the contents of Misc/gdbinit in the Python sources to ~/. gdbinit -- or copy it from Subversion.

Which Python is GDB using?

23.3 Extending GDB using Python

GDB can be built against either Python 2 or Python 3; which one you have depends on this configure-time option. Python scripts used by GDB should be installed in data-directory /python , where data-directory is the data directory as determined at GDB startup (see Data Files).