The scl enable ... command creates a new shell; you've probably ended up with several nested layers of shells if you've been testing this. If you exit, you'll probably see the gcc --version output from the base system. See how deeply-nested you are with something like pstree -s $$.

To run gcc --version with scl, just put the command on the scl line:

 scl enable devtoolset-9 'gcc --version'

Reference: The Red Hat Developer Toolset 9 User Guide (pdf).

I feel foolish because I figured it out right after posting, but I figured I'd share because it isn't intuitive and hopefully this will save someone some time.

The package you want to install is not gcc. You want to install gcc-toolset-9. You will notice after installation that you will still get 8.x if you do a gcc --version. gcc-toolset-9 installs to /opt. The idea is to give you a separate development environment. See this documentation. After installation you can run the updated gcc with /opt/rh/gcc-toolset-9/<username>/bin/gcc.

As @Stephen Kitt pointed out, you can actually get a shell with the updated toolset by running scl enable gcc-toolset-9 bash. Running this command will open a new bash session with the environment variables appropriately updated.

You can also run gcc v.9 directly with scl enable gcc-toolset-9 gcc <your_gcc_args>. See this documentation for details.