Explicitly mentioned is that all Zulip features are included in the free plan.
The self-hosted offering is notably described as 100% open source software in the tab heading above all the plans, paid or free. https://zulip.com/help/zulip-cloud-or-self-hosting confirms this interpretation. It’s as owned as any other open source software. https://zulip.com/self-hosting/ even confirms that the self-hosted offering is the same software as Zulip Cloud.
The mobile push notification service is also open source and can be self-hosted for free, although this requires recompiling the mobile apps with a different secret and distributing the modified apps to the desired mobile clients. Zulip has no way around this due to Google and Apple’s push notification security models.
You do have to pay for a server, but that's kind of inevitable (as in: a server is required to run most chat software, so someone will have to rent one).
I can only read this as an oxymoron