The original question is generic and the answers are specific to the presenting problem of cycling through tabs. I had this problem for a very different keyboard shortcut.
In my case, installing MacVim added a Service of “New MacVim Buffer With Selection” and I have had the keyboard shortcut of Ctrl+Shift+M set there for several years and I'm very used to using it.
But this one proprietary app was not accepting the Ctrl+Shift+M hotkey. I could change the hotkey to a wide variety of other key combinations, e.g., Ctrl+Shift+2 or Ctrl+Shift+, and it would work just fine. Something about this one hotkey was not working and the app has no mechanism for custom hotkeys.
The issue for me is that my fingers remember Ctrl+Shift+M and I didn't want to have to relearn that in all other contexts just to be able to use the command in this one proprietary app. I thought maybe I could somehow duplicate the Service and assign a different hotkey to each insance, but other threads convinced me that this was not a good option.
The solution from Darboscalante gave me the clue I needed.
In the Shortcuts window under App Shortcuts it’s possible to set an app-specific hotkey for any existing menu item in an app. And “New MacVim Buffer With Selection” does indeed appear in the apps Services menu. My own personal Services are there too. I added an App Shortcuts entry for the app, used the exact menu item name, set that to Ctrl+Shift+, (comma, next to the M), and voilá, both shortcuts would open a MacVim window containing the selection.
This is a more generic solution to the original problem: The Services menu can have one hotkey and the App Shortcuts can call the same Service with a different hotkey.
For my specific purpose I discovered that I could put Ctrl+Shift+M in the App Shortcuts so now I have the same shortcut in all contexts and my (different) problem is solved, too.
I hope this helps.