So I have Streamlight RAIL MOUNT HL-X LONG Gun on my 1301 and I put around 25 rounds of slugs through it today. When I got home I noticed that plastic on my light is melted a little and also notice that the Nordi Extension is also a little grey like it got hot.

Has anyone ever had this happen? Can anyone think of why this would have happened?

Usually on a SG the light is behind the muzzle enough it doesn't get in the way of the muzzle blast.  Yours must be mounted further forward.

I'm not sure about the particular light your using. I've seen a lot of Surefire lights on non-SGs with muzzle blast burns on the bezels and lens. Surefire uses glass and not plastic as the lens and  you have to wipe off the soot and keep it clean.

If the lens is melting maybe there is a replacement made of out of glass available.

Maybe it is just too far forward. I am totally new to shotguns. I have never had a issue with my lights on my AR or Scorpion doing this.  Side note I also lost the switch off of my hand guard that I had on there with the 3M Tape, I guess it couldn't handle the recoil.  I want to get a different hand guard, but impossible to find.  But this is why you should take your stuff to the range and find out what issues you may have.

The grey is carbon/lead residue. You can wipe that off with oil/CLP or use a pencil eraser. The plastic on the light more likely than not didn't melt but essentially got "sandblasted" by the particulates in burnt gunpowder.

This. I doubt you could possibly load the gun fast enough to fire it enough to actually melt the plastic, let alone harm a metal tube.

Fouling blast it? Absolutely. Kill it with shock/recoil? Possibly. Get the lens dirty enough it's basically useless without cleaning? Absolutely.

Wipe it off with your choice of oil on a rag and don't worry about it.

If you do kill it, streamlight will swap out a new one. I know for fact their handgun lights are not built for 12ga

This is why i use aluminum body lights on my shotguns so i can wipe them with solvent.