You want to use GNU Make, and then you can ignore the Make dependency tracking. GNU Make is much easier when you only use so-called phony targets (consult the manual), which always execute without doing any dependency tracking.

As for the advantage, a makefile will definitely perform both go build and docker push, rather than just (say) docker push, an ever-present risk if you have to rely on your fingers to type these things in, or rely on your eyes to check that you recalled the right command from the history. It will also explicitly tell you the build failed rather than relying on you to do echo $? or for the tools to have some obvious output in the error case.

A shell script is also an option. Makefiles have some helpful extra features: by default, commands are echoed; by default, the build fails if any command exits with a non-0 exit code (please consult local Unix person for the details); a Makefile inherently has multiple entry points; and, a Makefile can also be easier to get working on Windows than a shell script, though if you can demand that people run it from Git Bash then maybe there's not much in this.

If you're still not convinced: that's fine! This is not a sales pitch.

(I've more recently switching to using a do-everything Python script for this stuff, which is more annoying when it comes to invoking commands but has obvious advantages in terms of how easy it is to add extra logic and checks and some UI creature comforts and so on.)