* Linux is capable of running without a bootloader
Of course it can - it *is* a bootloader (not a generic one, but it bootloads itself)
Although in this case it isn't running without a bootloader.
* Linux is capable of depending (or at least being semi-dependent) on existing libraries
No it isn't. Linux doesn't link against any libraries. It doesn't even link against the standard C library.
* It is not selective to a particular console(s)
That's because it's a kernel and not an app. Apps depend on the system, kernels don't.
* Though in most cases Linux runs on a low level, defining it as a kernel, there are some ports which do not. For example; as a PSP developer, I know that making a kernel for PSP is impossible
No it isn't. Booting from that kernel on the metal may be impossible, but you can definitely run one emulated - which is what that video is doing.
Summing it up, though it is possible to emulate a pc on a psp, it is not possible to emulate ALL of the features of a pc on a psp. Linux would have no access to ME accelerated audio, memory access for flash0/flash1/flash2/umd, while running within the emulator. Additionally, the kernel would run freakishly slow unless it had direct access to SCE graphics functions since rendering unswizzled images to the screen reduces the framerate by about 75-80% (averaging about 60 fps without swizzling and 400+fps with swizzling and 600+fps if you also take advantage of offscreen memory). Altogether, my point is that somehow the Linux running in the video above has to somehow be accessing kernel mode functions.
It doesn't need to be emulating an x86 PC - which would be freakishly slow. It can just emulate the handful of hardware calls Linux needs, and the graphics driver could easily escape the VM and make native calls. This is what happens if you install Linux in something like VMWare and then install the VMWare tools. The driver jumps out of the VM and to the host to do the work.
In (for the love of god notice these words--->) MY OPINION, Linux can be considered as an application; a shell like DoorsCS.
Calling it an opinion doesn't make it valid. You are basically saying "it's my OPINION that 1 + 1 == 7" - it is factually wrong, period.
Appending on; the video clearly shows that the guy didn't run dosbox/bochs/whatever and that Ubunto called the PSP's built-in osk and dialog routines. Proof that it's using psp libraries.
Well, not necessarily. He could have re-created what the PSP's built in osk looks like, or he could have ripped that out of the PSP ROM, or he could have a driver that escapes his VM and invokes them. The Linux kernel itself isn't doing any of that, though.