Very unique has its place, if you consider two objects that are made up of parts, one object that has a previously unused combination can be said to be “unique.” However, if it is still similar to another, common, object, while it’s unique it shares some characteristics. Consider, now, and object that serves the same function but is vastly different in composition (different parts), we could call an object like that “very unique” in comparison to the “slightly unique” object 1, since object 2 has more unique parts and composition to fulfill its intended function than object 1.
Say you have some sort of Degnoming object, it has three parts, A,B,C. Now I come up with my own, unique, Degnoming utensil, having parts A,B,D, pretty much the same function, unique composition. Now you make your own Degnoming utensil, to shame mine and the commercial one, it uses parts X,Y,Z, similar effect, but different way going about it, and a much different composition, since it is more dissimilar, it is more “one of a kind.” Both my Degnomer and your Degnomer are unique, but since yours has more unique parts and functions, yours is “very unique” as opposed to my utensil which is just slightly unique.
If you refuse to accept this on the grounds that unique has little to do with the degree of disparity, note that Merriam Webster gives “unusual” as an alternate definition of unique.
ETA: OP, I’ve heard “I have that exact/precise lawnmower” before, if you swap “the” for “that” it’s not an odd phrase at all.