|Who are they?|
This is just a mere theory based on observation of the different games that contain the Happy Mask Salesman and Beedle which they seem to exist outside of a temporal correlation of time and space, or to use a better word atemporal beings, because no matter where you are, no matter where you go, you will always find them, up to the point I am thinking they could belong to a separate race of characters such as beings that do not follow the same physic rules as Hylians but also they could be guardians left by the Goddess's to protect the Hero named Link through the ages.
Beedle and the Happy Mask Salesman through their different incarnations exhibit similar abilities to each other, they know events that are happening around them, they have abilities to teleport around the field of gameplay, they seem to be immortal as enemies just ignore them or simply don't mess with them; another ability the two characters carry between each other is to simply have access to a large amount of objects and pull them out of nowhere to assist Link.
The knowledge of the Happy Mask Salesman seems to be vast and there are proof he traveled to different universes based on his first appearances where he keeps mementos int he form of masks through his travels on the different universes.
A theory I have, and I am sure many others had it also is that the Mask Salesman can alter his appearances to adapt to the timeline he is in, or at least to blend with the population which let people wonder who he really is? And if he could be related to his last incarnation that goes by the name Kilton.
Kilton and the Happy Mask Salesman have a lot in common besides their obssession with masks, they only appear at certain moments, have quirks on how they react to link but also seem to create powerful objects to assist the player. Kilton itself could be HMS who adapted his appearance to the unified timeline.
Since the unified timeline contains pieces of the previous ones, then it wouldn't be strange that HMS would find a way to survive but also to discard his previous appearance that was Hylian related into a more "creepy one" since there was no need to try to blend in with the decimated society that Hyrule turned to be after the Calamity Ganon.
Beedle on the other hand I find him far more interesting, because he has been since the beginning of time aiding Link as a shopkeeper; around Breath of the Wild he can travel everywhere in the blink of an eye or essentially be where and exists where you are at the same time.
For me Beedle is omnipotent and in a certain way omniscient such as the HMS but regards the incarnation his appearence always keep a similar look such as the bowl-do and the same mannerism over the different timelines and millennia, which let me think he could be the same person you find on the different placements.
The problem is to explain the existence of these two characters towards the multiverse theory, even on a epystemelogical way is difficult because it will lead these characters as an extension of a higher God or Goddess that is beyond the Golden Goddess's and that deity manifest itself in the most humanoid or monstrous form that is always behind Link to ensure his victory.
Some people say Zelda being the reincarnation as Hylia is the Goddess of Time have enough power, but on her humanoid form even if she wields great skills she seems to be more of a herald of the higher beings or a lesser god that rejected her divinity to become a guardian for eternity trapped in a cycle of reincarnation, while Beedle and HMS seem to defy every form of logic and manipulate time and space and time and seem to have an absolute power to manipulate their surroundings to always be there, vigilant.