I think it's relevant. I expect app designers to use the keybindings that make sense based on what the user wants to do instead of blindly adopting whatever guidelines the OS suggests.
If you're in the shell, "tab" should tab-complete, not move to the next terminal you have open.
If you're in a shooting game, "tab" should reload your weapon or show scores or whatever, not advance your cursor to the next target.
If you're in MS word, tab should indent the next line, not move to a different control.
If you're writing an email, same thing, because "writing" is most like "writing in MS word."
Re: your second point, Gmail is a really good "real" web client, at least for most people, and this is a small step away from that.
None of your other examples have the conflict of running within a browser, and the browser already having an expected behaviour for tab, including an expected behaviour for tab inside a text box / edit field.
If you're in the shell, "tab" should tab-complete, not move to the next terminal you have open.
If you're in a shooting game, "tab" should reload your weapon or show scores or whatever, not advance your cursor to the next target.
If you're in MS word, tab should indent the next line, not move to a different control.
If you're writing an email, same thing, because "writing" is most like "writing in MS word."
Re: your second point, Gmail is a really good "real" web client, at least for most people, and this is a small step away from that.