That is not global. From the uv getting started docs:
"When Python is installed by uv, it will not be available globally (i.e. via the python command). Support for this feature is planned for a future release. In the meantime, use uv run or create and activate a virtual environment to use python directly."
So yes, one needs mise/asdf/pyenv or similar for global installs for now.
Or via the global "python-preference" option set to "only-system".
Cf https://docs.astral.sh/uv/concepts/python-versions/#adjustin... and https://docs.astral.sh/uv/reference/settings/#python-prefere...