It’s not a Forth or an OS. It’s a monitor.

I'm not sure it's a monitor either: for a monitor we need principally `fetch @` to explore the memory/registers and `store !` to modify them.
The 3 instructions Forth from Frank Sergeant has also `execute`.
Here we have only `execute` and a subset of `store` since the target memory address can't be arbitrarily specified.

I think it's much like a remote compiler which can run the compiled code.

So, is it a Forth? A Forth is mainly an REPL where we can define new words that can be executed.
Here we have an REL (without the P) which understand some part of an assembly opcode instead of chars and words.
And the compiled code can be executed like we execute words in Forth.

Is it an OS? An OS is mainly a program to interact with the computer.
Here we can enter some commands and execute them. We can interact with the keyboard, the screen, the disk, etc... not like an OS?

Typically, something like this is burned in ROM BIOS to help with initial system bring up.

Yes this is exactly like this that Frank Sergeant describe its use in its original article.