Is the use of the Super Game Boy on the SNES allowed for this speedrun?

Absolutely, it's what a lot of runners who don't prefer emulators use.

But, is it being considered that Super Game Boy (1) runs faster then the actual gameboy and Super Gameboy 2?

Then I believe SGB1 is illegal

Does the fact that it is licensed by Nintendo change the legitimacy of it?

Well, I don't doubt the legitimacy of the hardware developed by Nintendo, but due to it, when it comes in terms of speedrunning, it's unfair that runners would be compared by RTA when one player on a specific hardware has the game running faster then others.

It's the same idea if someone on an emulator would overclock the game speed to 2x and be considered capable of finishing the game twice as fast.

Just for reference, Visual Boy Advance, the emulator, is also sometimes considered unusable do to the same reason SBG is: BGB and Gambatte being the better alternatives

It could be possible that people playing on different hardware would have a constant to which their final RTA time is multiplied, to reflect how much faster the hardware is running on; On this website, someone has done that, but the problem could be that the speed of the SGB depends on what game is being played, so it would be hard to believe that constant is acurate. https://kb.speeddemosarchive.com/Super_Game_Boy_timing

Looking at other GB leaderboards will probably tell you in the rules with gb emus are illegal

For GB games, SGB is generally allowed. Times are converted to the original clockspeed (according to the SDA article that Ravetuba posted. A web app for that purpose by Nudua can be found here: http://nudua.com/convert ). As far as we know, the framerate differences are consistent over all GB games. The emulators Gambatte and BGB are generally allowed for running GB games. Exceptions may exist, since there are parts of the emulation that might conflict with certain memory manipulations in glitched categories of certain games.

That said, and contrary to what Breadpan replied first, SGB is not used by many speedrunners (due to the inconvenience of having to convert times and the skipped frames and distorted sound that result from the inaccurate framerate). Nintendo released an updated version of the module which runs at the correct framerate of the original GB: the SGB2.