Yes, bootloaders and boot managers are started directly by the firmware.
(Although some firmwares are so complex that you could practically call UEFI an "OS", and its bootloaders "UEFI programs"...)
- Could Windows Boot Manager exist without Windows? Yes.
- Could it replace GRUB? Maybe, maybe not. (Although it depends on what OS you mean by "UNIX"!) It would be easier on UEFI than on BIOS.
- Could GRUB replace the Windows Boot manager? Maybe, maybe not. (Depends on the Windows version.)
The problem is that OS kernels have different ways of being started, and expect the bootloader to provide certain initial parameters, such as which disk to boot from, what "kernel commandline" to use, where the Linux initramfs is, and so on. For example, here's the Linux boot protocol, and here's the Multiboot spec used by some BSDs.
So you cannot tell Windows BOOTMGR to start
vmlinuz directly, and you cannot tell GRUB to start
However, sometimes that job isn't done by the boot manager itself, but instead by a small "stub" bootloader which can be started in a standard way. And that stub bootloader could be started by a different boot manager than usual.
For example, Windows BOOTMGR first starts
winload.efi, and that's where all the preparations for starting
ntoskrnl are done.
This means that you can make GRUB boot Windows by starting
winload.efi, without going through BOOTMGR.
Similarly, Linux kernels often come with a built-in "EFI stub", so that the kernel itself can be run as a standalone UEFI program.
So if your Linux kernel has its own "EFISTUB" option enabled, or the systemd-boot stub attached, then you can make Windows BOOTMGR start it directly without using GRUB or anything else.