For instance, let's say that I install Spotify with Homebrew. If a Spotify update comes out, will it also be available with brew upgrade the same way and day as if I were using the app installed via the Spotify website?
[...] will it also be available with brew upgrade the same way and day [...]
No, you will have to run in the command line:
brew upgrade
to upgrade and wait a few days that someone push an upgrade into https://github.com/Homebrew/homebrew-cask.