According to reddit (and partially the official manual):
Green is after story. Orange is for completing HR100, Blue is for entering MR, Purple is for clearing MR story.
MR200 is yellow
It's an indication of progress through quests (not through ranks like the reddit post suggests!). Each of the quests to unlock the colours are optional, and are unlocked once you have passed that point in the game. For example, the MR200 quest:
Apparently you need to do all the optionals that upgrade equipment like the mr 150 and mr 175 quests to upgrade mantles, to unlock it.
I've searched various forums in an attempt to find the relevant quests for each rank, and will link the source for each one in format:
[Color](source) - Description
Green - Beat the final story quest, Land of Convergence
Orange - Beat The Sapphire Star's Guidance
Blue - Buy the Iceborne DLC
Purple - Beat the final story quest of the DLC, Paean of Guidance
Gold/Yellow - Beat Master Hunter of the New World, unlocked by completing all the MR optionals
There is no indication that you can change this colour anywhere on the internet, though if you get two, please try, and tell me.