Currently, the 2.80 export scripts are NOT COMPATIBLE with 2.81! 

Well, they are mostly compatible, but animation exporting is completely broken. Something must've changed internally which breaks the information, creating weird results in GameMaker. I'm working on an updated script at the moment, it's just inconvenient right now.

The 2.80 and 2.79 scripts should still work fine in their respective Blender versions.

This also has an unfortunate consequence: It is very unlikely that I can back-port updates to older versions of Blender from here on. The existing scripts will remain, but the changes between Blender 2.79 and 2.80 were already a nuisance, but now that 2.81 is even more different and incompatible with the 2.80 script, back-porting will prove to be even more challenging, and unlikely to be worth the time.

Any questions about the update, backwards compatibility or anything else, leave them below!

A little update on this:

So, animations themselves are not broken - after extensive debugging, I found they are exactly the same. In fact, if you export the animations from 2.81 and apply them to a mesh exported from 2.80, it actually works fine! So this means something has instead changed with regards to calculating blend weights, but only when converting to left handed mode. A weird bug, but at least I now have a vague idea as to where the problem lies. 

It did actually result in me making a tiny optimisation for exporting quaterions to left-handed mode too, which is nice. 

The good news is that this slighlty increases the chances of me backporting some code again!

Another Update on this! 

I found the root cause - bone orientations get messed up on armatures when they get flipped in 2.81 and newer, which in turn breaks the animations (since they are all releative to the orientations).

I've reported it as a Blender bug and it seems to be confirmed! With any luck, this will be fixed and the script will work with newer versions again.

If this turns out to be a bug, instead of a permanent change, back-porting the code to other 2.8X versions will be fine again. There will still be some trickiness for 2.79, but it is still probably doable too.

One final update on this:

The bug has been Resolved by the Blender Developers, and I can confirm the fix will be in Blender 2.82!

However, unless the fix is added to 2.81 and 2.81 is updated with fixes, it will remain unsupported by the export script, which basically means the scripts will work, but armatures exported to left-handed space will be broken.