I've been seeing issues with Cmd-H ever since upgrading to Ventura, but unlike the Save As… dialog cursor bug, I haven't seen this one mentioned elsewhere. AE often fails to hide when I use Cmd-H. The UI focus shifts to the Finder, but the AE window remains visible. If I minimize AE, undocked panels remain visible and cause issues. Using the menu command After Effects > Hide After Effects seems to work more consistently than the keyboard shortcut, though that may be coincidence. The problem as a whole is intermittent, not constant, so it's hard to test and provide specific detailed descriptions.
Thanks very much for reporting this issue. When you're in the state where After Effects does not hide correctly, try clicking on the After Effects icon in the Dock. Does it hide as expected after that? It sounds like After Effects might be in a "half-focused" state to start with and clicking the Dock icon might fully focus it so that it can be properly hidden/defocused.
Additionally, what is "Use System Shortcuts" set to in Preferences > General?