The original poster did not mention what type of computer or operating system he was using while attempting to install TensorFlow alongside Python 3.9. The error could be linked to working on a 64-bit Mac with the M1 chip (I recently experienced the same error described above while working on a Mac M1 in a Miniconda environment with Python 3.9.13). I solved the error by running
python3 -m pip install tensorflow-macos
from Terminal (in the Miniconda environment). TensorFlow installed normally alongside Python 3.9.13.
I do recommend installing Miniconda (or Anaconda as others have suggested), because it will allow you to easily create development environments with whatever version of Python modules or dependencies you require at the moment. See https://docs.conda.io/en/latest/miniconda.html. The larger Anaconda comes with a user-friendly 'Navigator' GUI which enables you to choose which environment is used to open a Jupyter notebook or other development environment, several of which come with Anaconda. See https://docs.anaconda.com/anaconda/install/