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)
Given that this is a lore question, and not a specifically 5e one, I would strongly recommend the 2e product Forgotten Realms Adventures, as it has a section on 24 cities of the Heartlands with exactly the kind of information you are asking for. The catch with it is that a lot of the listed characters there are now dead (if you are playing past the Second Sundering). But even then, in terms of the social structures, most of the data are still remarkably intact. For each city in the FRA, the following information is provided: Who rules, who really rules behind the scenes, population, major products, armed forces, notable wizards, notable churches, notable rogues' and thieves' guilds, equipment shops, adventurer's quarters (lodging), important characters, important features in town, local lore and finally a small map.
As for the 3e, FR Campaign Setting (FRCS) is excellent. At 320 pages and with a small font, it is one of the most information packed D&D books ever published. (It received the Origins Award for Best Role-Playing Game Supplement of 2001.) It provides a decent amount of information on the cities, but not in the itemized format of FRA. Instead it has data on regions/countries and cities are given as individual paragraphs. So it covers a larger geography (significantly larger than FRA, or 5e Sword Coast Adventurer's Guide), but provides somewhat less detailed data on particular cities. One excellent thing in FRCS3e is a map of trade between regions.
The 4e Forgotten Realms Campaign Guide (FRCG) is structured in the same way as FRCS. However it lacks the level of detail FRCS provides. The 4e "Points of Light" philosophy is arguably reflected onto FRCG and the particular data on the cities are very limited. Moreover, 5e D&D (and the in-game mechanics of the Second Sundering) has mostly returned the setting to pretty much how it was in 2e/3e.
Finally, the 5e sourcebook Sword Coast Adventurer's Guide (SCAG) has partially returned to the style of FRA (cities are discussed explicitly), with a mix of FRCS3e. It covers more limited geography than FRCS3e, and the data on the cities is less organized than FRA (and some data are missing). Its main pro is that it happens to be the most recent published material. If one did not care about editions, FRCS3e and FRA are more suited to answer the question you have asked.
PS: Both FRA and FRCS3e are available as watermarked pdf downloads that you can buy on the web for less than half the price of SCAG.
Best Answer
While this question has been adequately answered, I recently spotted a source that should confirm reliably that Abeir was never the name of a separate world prior to D&D 4th edition.
In Dragon magazine 359 (September 2007), page 28, in the article "Unsolved Mysteries of D&D":
"Ed" here is Ed Greenwood, perhaps foreshadowing the sundering of Abeir-Toril.
Certainly, if the experts of Dragon magazine did not know what Abeir was by the end of D&D third edition, nobody knew.