I use a great app called magnet that allows me to use keyboard shortcuts to resize windows to certain portions of of the screen. I would like to remove its icon from the menu bar because my menu bar is already very cluttered. But I still need this app to be running at all times. (Yes, I've checked the app's settings; there's no option to remove the menu bar icon.)
Bartender will let you organize and hide icons on your menu bar. This is probably the best-known solution.
Another alternative that you might want to check out is Vanilla.
I use Bartender and highly recommend it, but it’s worth downloading both and checking them both out.
Before you ask, no, there is no way to do this without 3rd party software, unless that feature is added by the developer.