The tmux.conf had this line:
bind -n Z kill-session
so tmux was just doing what I was telling it to do.
As @KamilMaciorowski suggested, tmux list-keys -T root | grep <the keybinding that is weird> is helpful in diagnosing this sort of thing.
Thanks to @Kamil Maciorowski, I have a solution. Three steps:
Define a master service, master.service, as following:
Description=tmux master service
ExecStart=/usr/bin/tmux new-session -s master -d
ExecStop=/usr/bin/tmux kill-session -t master
Define the first service, test1....