As there is currently no linux package for the CLI, I use the github release.
I use eget to install and manage binaries installed from github.
This could be useful for others as well. I installed it using the following config:
# ~/.eget.toml
["stackitcloud/stackit-cli"]
asset_filters=["stackit-cli_", "_linux_amd64.tar.gz"]
eget stackitcloud/stackit-cli