Is there a way to see the application switcher (the one pops up when hitting
CMD+TAB) on all connected displays?
To my knowledge, you can't make the default application switcher do that. 3rd party apps definitely can... but many don't seem to do it.
The only third party app I know that does this is: Contexts (~$10), but it's also pretty different from the default app switcher.
Another one I wanted to mention is Witch (~$14), that is closer to looking like the native application switcher, but isn't able to show up in all of your screens. It can however be shown in the screen with the active window or in the same screen with your mouse pointer.
Both of these have trial versions available.
The big thing for me with these third party application switchers is that it is possible to hide applications from them.
For example... if you use Path Finder, you may want to hide Finder, so you don't accidentally open it. Or if an application just doesn't have the "hide dock icon" option, meaning that it will show up in the application switcher by default, you can hide it in the third party application switcher. Many reasons for why you'd want to hide an app from the application switcher.
I personally use Witch and Ubar, which is a Dock replacement that also can hide applications... Contexts comes with its own dock replacement (sort of). As a heavy Spaces user, I really like how the "Sidebar", as it's called in Contexts, can divide the open apps into groups by spaces. Makes it pretty nice and organized.
The thing I could never get used to with Contexts was the actual application switcher and its vertical list style (with no option to make it horizontal). I spent way more time looking at the list trying to find the right app compared to the native horizontal style, which is not available in Contexts.
As a bonus I just wanted to mention HyperSwitch, which mainly brings really good previews to either its own custom app switcher or extends the native app switcher to show a preview, but as far as this question goes, it will not extend the application switcher to multiple screens.