They keep referring to Reach/4 as being done on 'their previous engine' so there's a good chance the only thing the Destiny engine shares with Halo is the tag system (which is generic enough that you could have it in any engine).
The only interesting thing is if Destiny was on an evolved Reach overhaul then this would be the first time two games (again, if Destiny is an FPS) on the Blam! engine are competing directly with each other. That's more nerd trivia than a fact anyone will actually care about tho
You'd also tend to see a decent amount of employee flow between 343 and Bungie as once you'd learn the engine, you'd have a foot in the door at both of them.
edit: I sometimes wish ODST had come out after Reach on the Reach engine. Reach and ODST are really the only time a Halo game has been co-developed with the full knowledge of what the other one would be. In my opinion ODST and Reach are developmentally/thematically an extension of each other.
edit2: in other words, they work as an expansion pack of each other more than ODST does as an expansion of Halo 3.
