Answering this because I'm not satisfied with the other answers.
Short answer: It's unlikely that Volcarona was intended to be a Legendary Pokemon, and, while it's possible that it was intended to be but changed for one reason or another, it's more likely that people may have thought of it as such because it has multiple characteristics that, together, are generally reserved for Legendary Pokemon.
Long answer:
The distinction that certain Pokemon are "Legendary" is relatively arbitrary. Most are referred to as legendary because official media refers to them as such. That being said, there are certain traits that many of them share, so we can try to define it. Legendary Pokemon tend to share a few characteristics: for the most part, you can only obtain one of each in a save file, most do not evolve, with the first evolving Legendary Pokemon being introduced in Gen VII, and only three lines of evolving Legendary Pokemon exist (Cosmog/Cosmoem/Solgaleo/Lunala, Type: Null/Silvally, and Kubfu/Urshifu), many are genderless and cannot breed, typically appear at the end of their region's Pokedex, typically have high base stat totals, and none are random wild encounters, instead having special methods of obtaining (if you leave a comment about roaming Pokemon, I will mail you a pipe bomb. That is a special method of obtaining). I will say that I'm not the most familiar with anything beyond Gen VI, so feel free to point out inconsistencies. Some of the criteria from this list were sourced from this Bulbapedia page.
The only one of these characteristics that Volcarona had when it was introduced in Black and White was the ability to evolve and breed, with the fact that it can only be obtained at a high level in a battle that you initiate by interacting with it in the overworld (a method also used by Legendary Pokemon, more often than not. Most other Pokemon that you initiate battles with by interacting with them in the overworld can be also be found in normal wild battles) perhaps being the most glaring reason people may have assumed that Volcarona is a legendary Pokemon. In fact, it also has a signature move, Firey Dance, something Legendary Pokemon also often have (yes, I know that many other Pokemon have signature moves, though it is infrequent that non-legendary Pokemon keep the exclusivity of their signature moves). In Gen V, the only other way to obtain it, other than through breeding, was to hatch the egg you obtain on Route 18 in Black and White, which is a pretty late game location if my memory is correct. Also worth noting that the only trainer that has a Volcarona in B/W is Alder, the Pokemon League Champion, and in B2/W2 the only trainer to have one other than Alder is his son, Benga. With all this said, it wouldn't be hard to assume that Volcarona is a Legendary Pokemon, especially if you're a 10 year old. It also looks badass, a trait that was shared by all Legendary Pokemon until Enamorous happened (and, if you're a 10 year old boy, the only outlier would be Cresselia, because pink is for girls, of course).
As a side note, Rotom and Zoroark were also treated in similar ways (if my memory is correct) for similar reasons, most notably that both can only be obtained once in battles initiated through overworld interaction with the Pokemon, and, in Zororark's case, it had a whole movie centered around it and was impossible to obtain it in Black/White if you didn't get a limited time event. Still salty about that.
