Browser (and other programs?) opens in the current X session the first time, then on the same session even if launched from another X session

Please, correct me if the wording of the title is wrong, as I do want to understand the correct terminology too. The think is, I can swtich using Ctrl+Alt+F _N_ between different window managers each launched on a different virtual console. I've always used one, but these days I'm experimenting with a new window manager (XMonad) while still mainly using the one I've used for years (i3), so I'm frequently switching from one to another and viceversa. In practical terms, I have this config in place:

[[ "$XDG_VTNR" -eq 1 ]] && exec i3
[[ "$XDG_VTNR" -eq 2 ]] && exec xmonad

One thing I've noticed is that if I open Chromium, the first time it will open in the current session, e.g. i3, but if open another Chromium process, it will open in i3 even if in the meanwhile I switched to xmonad. Same thing the other way around. Do I have control on this behavior? How?