That used to be a very popular naming convention in the 80s and 90s, when "open standards"/"open systems" was a big craze. The term "open source" was only coined in 1998 (or maybe late 1997?), and much of the term's adoption didn't happen until the first decade of this century.

Proprietary operating systems which added POSIX/Unix compatibility frequently used "Open" as branding, either for that compatibility support or for the OS overall. As well as DEC/Compaq/HP(E)/VSI OpenVMS, there is also OpenEdition, which is the IBM's original name for the Unix compatibility component of their MVS mainframe operating system–MVS was later renamed OS/390 and now z/OS (although to be pedantic about it, OS/390 and z/OS refer to the operating system as a whole, officially MVS is just certain of its core components, and as such MVS still exists today as part of z/OS), while OpenEdition was renamed to Unix System Services (sometimes abbreviated to USS, even by IBM, although purists object to that, since USS was already in use as the abbreviation for the Unformatted System Services component of IBM's VTAM mainframe networking package.) Due to this Unix compatibility component, z/OS is actually a certified Unix (UNIX 95). It is also somewhat more than just a compatibility component, in that many newer OS services require it to run – basically anything written in Java, or anything which is shared code with non-mainframe IBM platforms, runs under it. A lot of the systems management frameworks in newer z/OS releases can't run without it.

There is also a POSIX compatibility component of IBM's VM/CMS operating system (now known as z/VM) - originally called OpenEdition too, later renamed to OpenExtensions. OpenEdition/VM was always significantly more limited than OpenEdition/MVS, in part because Unix APIs were technically more challenging to implement under VM/CMS than under MVS (since native VM/CMS is further away from Unix than native MVS is), in part because less was invested due to less market demand at the time (and very little remaining today). OpenExtensions is basically in deep maintenance mode, IBM still supports it but have publicly said they have no plans to add new features; OpenEdition is in somewhat of a shallower maintenance mode – they aren't interested in updating it from UNIX 95 to a newer UNIX standards release (UNIX 98 or UNIX 03 or UNIX V7), but given they actively use it to provide higher level OS functions (such as Java support), inevitably some enhancements sometimes get made to support new requirements in those areas. Generally, for anything more demanding, IBM will recommend using z/Linux (Linux on the mainframe), or zCX (a port of Docker to z/OS, wherein z/Linux is run under a hypervisor which is built in to the z/OS operating system.)

Similarly, when Siemens added POSIX compatibility to their BS2000 mainframe operating system (mainly used in Germany), they renamed it to BS2000/OSD (OSD = Open Server Dimension). BS2000 is still around, although Siemens ended up selling it to Fujitsu.

Fujitsu owns at least three different mainframe lines – as well as BS2000, they also bought the British firm ICL who owned the VME mainframe platform, and they have also have a "clone" of IBM MVS, MSP. The "clone" is not really a clean clone, they actually stole a lot of IBM's code to build it. IBM sued them (and Hitachi too who were doing the same thing), and the lawsuit settlement included a license from IBM for them to legally continue doing it (for which Fujitsu and Hitachi paid IBM several hundred million dollars each). MSP is less than 100% compatible with MVS, and only with old 1980s/1990s vintage 31-bit MVS. I believe it currently only survives in Japan, although it used to be popular in some other countries too, in particular Australia. A lot of Amdahl machines ran Fujitsu MSP, and then Fujitsu bought Amdahl. But, back to the topic of Open – when ICL added a POSIX compatibility subsystem to VME, they renamed it to OpenVME. And OpenVME is still around too; its major remaining use is inside the UK government, which has various core systems trapped on it.