Linux

Forking Linux would involve taking on a huge maintenance burden, so everyone just uses it, though often basing on an older version and/or with some custom patches. That is typically how healthy open source stuff works.

Companies DO put brand names on systems built on top of Linux (or a BSD) all the time though, often ones that don't make it obvious that's what it is. ChromeOS and Android are both Linux based, but Android doesn't ship most of the UNIX-y parts that are typically layered on top, and instead uses their own (also largely open source) components. ChromeOS is actually a fairly close relative of Gentoo with a few custom pieces.

Google has their own internal project for a kernel called Fuchsia, and it's really interesting modern OS development that they've assembled a bunch of experts to work on... But it's increasingly unclear if they plan to deploy it on customer facing products.

A ton of appliance type devices are basically very tiny custom Linux systems, often assembled with tools like Yocto. A lot if the vendors who sell components to go into said devices contribute code and/or money to Linux and Yocto, in order to make their products more attractive to device builders and avoid having to make and maintain their own tooling.

Most consumer routers are basically Linux (usually with a minimal userland like BusyBox), often essentially shitty old customized versions of OpenWRT. Sony alpha cameras? Customized Linux. Off on the BSD side, CellOS and OrbisOS that the PS3 and PS4 run, respectively are modified FreeBSD. Open Source OSes and tooling are everywhere because making, maintaining, and building tooling and developer support for an OS that runs on especially relatively large modern computers is a big, hard project, so very few entities try to do it themselves.

As I see it their options are:

1. Copy it without meaningfully changing anything, i.e. just redistributing Debian or Ubuntu with some logos and desktop backgrounds - there is no reason to install this on your own and no one will care. This is effectively the same thing as customized Windows installs that they ship.

2. Creating a custom Linux distribution. This is feasible and has already been done (System76 created Pop_OS! based on Ubuntu; Android and ChomeOS are essentially new Linux distributions built from the ground up, taking only the kernel and ignoring the existing ecosystem), but requires serious maintenance work to be any good and offer real advantages over existing distros.

3. Forking a kernel like Apple did. This has fairly limited purpose from the perspective of laptop and workstation OEMs that use regular off-the-shelf parts. This is usually done to provide support for custom hardware (Apple) or for proprietary software that requires deep integration with the kernel (VMware).

No one will think of option 1 as a serious competitor in the OS market, option 2 requires a ton of work and motivation, and option 3 is useless for these OEMs. Software just isn't their business and a cheap copy offers no real advantages over shipping an existing thing.