Is the Linux system clipboard represented in the file system somewhere as a device?

AFAIK no.

But you can use xclip if you want command-line access to the X11 Clipboard

No..

The operating system is not for GUI/Application layer semantics it only provides the raw abstraction to present a consistent, pretty system to user-space applications. If you want to do something like this, I would advise you write a system daemon that applications can use as a copy store and access through system IPC such as DBus.

Standards in the freedesktop.org standards may define standards for GUI interoperability and advise they communicate through something like DBus.

Rather than a kernel space system, you may want to manage copy and paste semantics above OS services such as IPC and keep the policy in user-land but through operating system mechanics.

Whilst a device driver presentation kind-of makes sense, IMHO it belongs in user-space as some kind of mini-database with source/target data and meta-data relating to encoding and so on ... none of which are strictly kernel concerns.

Please don't write a copy/paste device driver :)

edit toned down the bolding ..

There is no kernel-level "clipboard" - it's a concept belonging to higher layers, such as X11 for example. Of course, nothing stops you from writing a device driver, user-space filesystem, or whatever, to make it visible in those terms!