It certainly appears not. They're using curl and reporting a UID they generate.
A Homebrew analytics user ID e.g. 1BAB65CC-FE7F-4D8C-AB45-B7DB5A6BA9CB. This
is generated by uuidgen and stored in ~/.homebrew_analytics_user_uuid. This
does not allow us to track individual users but does enable us to accurately
measure user counts vs. event counts
You can always even check the code that’s running on your own computer. If you installed Homebrew in /usr/local the relevant files are /usr/local/Library/Homebrew/utils/analytics.sh and /usr/local/Library/Homebrew/utils/analytics.rb.
Of course, you have to trust homebrew, but hell, you're already running their software, so it's a little late.