Dropbear does not register user sessions via PAM/dbus
Probably it is expected due to limited PAM support in Dropbear, but I didn't want to miss reporting it, just in case it is easy to implement without causing much overhead and loosing the light nature of Dropbear.
In a Dropbear SSH session on a systemd-driven system, DBUS_SESSION_BUS_ADDRESS is not set, which allows to run systemd services in user mode via systemctl --user .... I'm not sure about details, but it's the libpam-systemd > dbus > systemd-logind stack which normally enables this, so I guess it means to implement support for the pam_systemd module.
This workaround from https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=751636#20 worked for me:
apt install psmisc
cat > /etc/systemd/system/dropbear-cleanup.service <<EOF
[Unit]
Description=Dropbear session cleanup
After=network.target
[Service]
TimeoutStartSec=0
Type=oneshot
RemainAfterExit=yes
ExecStart=/bin/true
ExecStop=/usr/bin/killall dropbear
[Install]
WantedBy=multi-user.target
EOF
systemctl start dropbear-cleanup && systemctl enable dropbear-cleanup
This solves non-terminated SSH sessions on shutdown (which is the same with OpenSSH), but does not fix starting Dropbear via systemctl --user 😉.