Why is the -s or -g option required while setting escape-time option in tmux?

Question

I am unable to understand why exactly the -s or -g option is required while setting the escape-time option in tmux. Here are some answers that seem to use the -s or -g option while setting escape-time:

• https://superuser.com/a/252717/115290
• https://unix.stackexchange.com/a/25638

Here are some experiments:

tmux kill-server
tmux
tmux detach
tmux

# Now we have two tmux sessions named [0] and [1] with [1] being the
# current session.

tmux set escape-time 2000
vim

# Press 'i' to get into insert mode.
# Then press 'Escape' to return to command mode. It takes 2 seconds.
# Enter ':q' to quit Vim.

tmux switch-client -t 0
vim

# Press 'i' to get into insert mode.
# Then press 'Escape' to return to command mode. It takes 2 seconds.
# Enter ':q' to quit Vim.

The conclusion of the above experiment is that escape-time option set in one session affects all sessions automatically.

Why is the -s or -g option required then?