All mentions, that I am aware of, regarding Bladesingers are below:

• AD&D 2^nd Edition / Complete Book of Elves

  Distinctive Appearance. Not only are they easily identifiable by their weapon of choice and catlike grace, Bladesingers are decorated with their weapon guild's distinctive tattoo. Each guild has a separate and unique tattoo depicting its style of weapon through an animal representation. Long swords are often represented by great cats such as lions or panthers, a whip would be shown as a striking snake, and so forth.

• AD&D 2^nd Edition / Elves of Evermeet

  On Evermeet they are organized into small warrior lodges, each specializing in a different weapon. They often tattoo themselves with old runes, and each lodge has a special symbol it's members tattoo themselves with to identify their membership.

Bladesingers are mentioned in the edition after AD&D 2^nd Edition, and despite its contradiction to 'guilds' or 'schools' there is no mention of any particular weapon (besides blades) to be in use.

• D&D 3^rd Edition / Races of Faerun

  Most bladesingers work alone, sufficient unto themselves, but in larger communities they sometimes have the opportunity to fight together in the same combat. Bladesingers are normally trained singly by another bladesinger, and the concept of anything as formalized as a bladesinger school is an absurd notion to them.

What that means, in that edition, is a school representing animals went to the way side. in the revised edition, D&D 3.5 Edition, Bladesingers were mentioned as well.

• D&D 3.5 Edition / Complete Warrior

  Bladesingers are elves who have blended art, swordplay, and arcane magic into a harmonious whole.

It is essentially a copy paste, with a lot omitted, from Races of Faerun. Only "swordplay" and "any martial weapon" is mentioned. Nothing about schools guilds at all. There is another book the Bladesinger is mentioned, but its essentially only a blurb.

• D&D 3.5 Edition / Races of the Wild

  Bladesinger (Complete Warrior): Practitioner of the quintessential elven martial art, the bladesinger is both a skilled mage and a deadly swordmaster.

Once again, only sword is mentioned; probably due to racial sword proficiency - and the implication of blade in the name Bladesinger. Eventually, near the end of this edition, Bladesingers became Duskblades, and lost its 'elf-only' motif.

• D&D 3.5 Edition / Player's Handbook II

  While the ability to cast arcane spells in armor originated with the elves, over the millennia the secrets of the duskblade have been disseminated to the other races, and today members of any race can become a duskblade.

No mention of guilds or schools, or even preferred weapons were mentioned. Any martial weapon it appeared to be - but spending a feat to use a whip...most people would say, "no thank you." Bladesingers were included in the edition after this one, much to my surprise. But, it discredits the 'different weapons' aspect entirely.

• D&D 4^th Edition / Neverwinter Campaign Guide

  Benefit: Choose a one-handed melee weapon with which you have proficiency, and that is a light blade or a heavy blade.

Now we come to D&D 5^th Edition, and you already have the published information on them thus far. Since, to my knowledge, that edition is to harken back to the days of simpler game-play and more intrinsic role-play; rather than the vice versa, take this opportunity to "Create Your Own" schools, guilds, and organizations.

5E Realms lore does not specifically address this. 4E Realms lore, on the other hand, does. Their exact origin is subject to some debate, even among the elves.

Collectively, the elves are known as the Tel’Quessir⁶ ("The People"), a title that encompasses Eladrin, Wood Elves, High Elves, Sea Elves, and so on. It is 'widely accepted' that the Tel’Quessir are native to the Feywild¹. It is believed that they first immigrated to Abeir-Toril over a hundred millenia ago, while the exact date is uncertain it is believed to have been prior to −24,000 DR, with some manuscripts indicating it was as early as −30,000 DR, the same time in which dragons were setting up their empires (Note: Source on this is from AD&D)⁴. The Wild Elves were the first to arrive.¹

The exact origin of the Elvish race is subject to some debate. Some ancient manuscripts suggest that they were the result of a battle between Gruumsh and Correlon that occurred in the Feywild. During said battle, Correlon was injured and his shed blood became the Eladrin.². Other types of elf, not liking the superiority this implies in that the Eladrin are the first and 'purest' of elves, disagree with this theory³. Another theory postulated is that since the Feywild tends to be a 'reflection' of the Prime Material Plane, and thus produces creatures that are echoes of creatures from the Prime, it is also possible that the Elves are one such echo... though what, exactly, they would be an echo of is uncertain¹.

Perhaps also worthy of mentioning is the current in-lore explanation for how the Feywild (also known as Faerie) is treated according to current lore. The Feywild was created as it exists today, a 'lighter' echo of the Prime Material Plane. It was made by Primordials in the far distant past, before recorded history begins.⁵

For a while, the Feywild existed alongside the Prime and interaction between the two was common. But, at some point in ancient history, the two drifted apart and travel between them became increasingly difficult². This is the explanation for why, in prior editions of D&D, the Feywild didn't "exist" in the Forgotten Realms. It was out there, you just couldn't get to it.

The Spellplague (the event that 'kicks off' 4E in the Realms) changed that, reshuffling the cosmology to drag the Feywild back into close proximity, and placing it in the accessible location it resides in today⁵.

Checking through published 5E Realms material, it does not contradict any of this... the 5E Realms cosmology is essentially the same as the 4E Realms cosmology, and the lore established within 4E is considered to hold true in 5E, except where directly overridden. And even then, the changes are usually justified in-lore (such as the array of gods and goddesses getting rejiggered due to Ao meddling with them).

The primary lore sources we have in 5E Realms are the Sword Coast Adventurer's Guide, the DMG, and published adventures. None of these go into detail on the origins of the Elves, or even on the specific nature of the Feywild... thus we may assume the lore of older sources hold true.

Sources

¹ Brian R. James and Ed Greenwood (September, 2007). The Grand History of the Realms. (Wizards of the Coast)

²Bruce R. Cordell, Ed Greenwood, Chris Sims (August 2008). Forgotten Realms Campaign Guide. (Wizards of the Coast)

³James Wyatt (December 2007). Dragon 361: A Fractured Family. Wizards of the Coast.

⁴Steven E. Schend and Kevin Melka (1998). Cormanthyr: Empire of the Elves. (TSR, Inc) NOTE: AD&D source

⁵Dungeon Master's Guide 4th edition. (Wizards of the Coast)

⁶Kim Mohan ed. (2015). Sword Coast Adventurer's Guide. (Wizards of the Coast)