Could you share the two URLs (yours and theirs)? It might have to do with the quality/formatting of the content. Have you checked their backlinks? Maybe they have lots of links pointing to that site from high-quality domains.
But you might be onto something with the idea that their entire site is dedicated to this single keyword.
Big caveat: Positions 1 vs. Positions 3 is really not a huge difference, and you will likely jockey for positions 1 – 5 naturally over time.