Zulip self-hosted is billed monthly. Still a form of rent. You don't own the version you bought perpetually.

According to https://zulip.com/plans/#self-hosted, the only things you get by upgrading from free self-hosted (which is absolutely offered) to paid self-hosted is to remove the limits on mobile notifications, which is a service that Zulip as an organization has to run and which therefore has an inherent cost, plus access to various forms of customer support.

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.