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Doesn't record a video, but I've been a happy user of LICEcap to make animated gifs of a portion of the screen that I share with co-workers (to github, slack, etc.)

Yes, Cmd+Shift+5 and “record selected portion”.

I think the goal was to record just the window, rather than a region of the screen.

That's the "Capture Selected Window" option.

but specifically for video it doesn't exist

Is there any trick to record a video of a given app window only

Quicktime Player does this.

New Screen Recording → Capture Selected Window

note that this is just a proxy for Cmd-Shift-5, and only takes a screenshot (not a recording)

Not a trick per se, but I use Giphy Capture for this kind of thing.