I guess you use a docker image from timescale/timescaledb. The Dockerfile used to build these images is public in GH repo timescale/timescaledb-docker. TimescaleDB docker image is based from postgres-alpine and thus inherits entrypoint from there. The description of postgres image presents available environment variables, which can be used during new container start. It includes POSTGRES_DB, which can be used to specify the database to create on start. If the available options is not enough, the entrypoint can be redefined by using option --entrypoint for docker run command.

Summary:

• use postgres image environment variables to provide your options, since timescaledb image inherits entrypoint from Postgres-alpine.
• provide custom entrypoint with docker run --entrypoint, see the doc.
• the dockerfile of TimescaleDB is available public.

Note that Timescale also provides docker image of TimescaleDB with HA for k8s. This one doesn't inherit from postgres and contains own entrypoint.