In case someone else runs into it, the noip.conf generated template from first-run (for me at least) ended up generating a misleading set of instructions. Above the interval line, the examples had a space between the value and the unit of measurement. At the time, there wasn't an explicit version number available, but as of Dec-13-2019, the Docker definition was last updated "A Year Ago."
# Examples: 5 m, 5 h, 5 d. Minimum is 5 minutes.
INTERVAL='2 h'
Including said space results in the container refusing to remain running, with the following being logged.
INTERVAL must be a number followed by m, h, or d. Example: 5m
run-parts: /etc/run_once/parse_config_file: exit status 2
Run-once scripts failed. Stopping container
Shutting down container...
fail: noip: runsv not running
sh: you need to specify whom to kill
Removing the space (between the value and the unit of measure) from the interval resolved my issue, and updates claim to be trying to happen for me. It looks like I still will need to click on the email link every few weeks to prevent the host from expiring, but that's from me being cheap and using the free tier of no-ip.
# Examples: 5 m, 5 h, 5 d. Minimum is 5 minutes.
INTERVAL='2h'