Oh, if you need to enter a user password in a unix prompt, you usually don't see anything (no asterisks *** or anything like that).
That's normal behaviour. It is still picking up the characters you're typing and you can even backspace-delete characters if you notice you mistyped.

As far as I know, there's no way to install the Homebrew package manager without root.

But if your goal was to install Bitcoin Core, I'd recommend pulling the source through git (like you've done) and building it locally, anyway.