Having outputs for nix-shell environment is pretty much meaningless β nix-shell does not build the outputs, it just sets environment variables from the derivation and then runs $stdenv/setup. So adding outputs attribute would just populate the output variables with output paths as described in the manual. But those paths would not exist in the Nix store unless you previously built the derivation used for the shell.
installPhase would not be invoked by nix-shell either.
If you want to specify what outputs of derivations listed in paths to link to the env symlink tree, you should use the extraOutputsToInstall argument of buildEnv: