In the snippet you posted, every appearance of $SESSION_NAME is expanded by the shell and each tmux gets the expanded value as a command line argument.
(Even if it didn't and you wanted to expand this $SESSION_NAME later inside tmux or inside a shell inside tmux, your export SESSION_NAME="my-project-name" may be futile because of how tmux builds the environment. The update-environment option or new-session -e … command may be useful.)
This includes $SESSION_NAME in bind-key -n C-c kill-session -t "$SESSION_NAME". Your tmux executes:
bind-key -n C-c kill-session -t my-project-name
Then for a different project the tmux command is like:
bind-key -n C-c kill-session -t different-name
The important thing is bind-key itself does not target a single session. bind-key -n C-c … changes the behavior of Ctrl+c for the entire tmux server. In your case the last bind-key overwrites the previous assignment and then Ctrl+c in any session will kill-session -t different-name.
I guess you could use the key-table option to set the default key table to something other than root, separately for each session. In each new key table you could set a different binding for C-c. I haven't tested this, I think it may be possible.
If I were you, I would do a simpler thing. I would do this:
tmux bind-key -n C-c kill-session
kill-session without -t targets the current session, so this binding should do what you want for my-project-name, different-name or any other session.
Frankly "any other session" is too much for me. I wouldn't want Ctrl+c to kill everything in my general purpose session(s). I would solve this:
either by leaving C-c alone and picking another keystroke as "global SIGINT", then advertently using one or the other;
or by explicitly "marking" sessions that should use Ctrl+c as "global SIGINT". It may be something like this:
tmux new-session -e 'TMUX_GLOBAL_SIGINT=1' \; \ bind-key -n C-c if-shell '[ -n "$TMUX_GLOBAL_SIGINT" ]' kill-session 'send-keys C-c'
Now in every session that has nonempty TMUX_GLOBAL_SIGINT variable in its environment Ctrl+c will kill the session. In every session where TMUX_GLOBAL_SIGINT is empty or not set Ctrl+c will work as usual.
Notes:
For an already started session you can change the setting with tmux set-environment -u TMUX_GLOBAL_SIGINT or tmux set-environment TMUX_GLOBAL_SIGINT 1 (use -t to target a specific session, not necessarily the current one).
A shell (or another program) started in tmux may inherit TMUX_GLOBAL_SIGINT. If it runs tmux new-session … then (depending on update-environment and such) the new session may also inherit the variable, even without explicit -e 'TMUX_GLOBAL_SIGINT=1'. Read man 1 tmux and learn about how tmux manages the environment.
