The two spaces Rx(RxR) and (RxR)xR have many isomorphisms between them (every possible coordinate change, for instance). But what matters, what makes them "canonically isomorphic", is not isomorphisms in how they are _constructed_ but in how they are _used_. When you write an element as (a,b,c), what you mean is that when you are asked for the values of three possible projections you will answer with the values a, b, and c. Regardless of how you define your product spaces, the product of (a), (b), and (c) are going to produce three projections that give the same answers when they are used (if not, you built them wrong). Hence they are indistinguishable, hence canonically isomorphic.
This is exactly the way that physics always treats coordinates: sure, you can write down a function like V(x), but it's really a function from "points" in x to "points" in V, which happens to be temporarily written in terms of a coordinate system on x and a coordinate system on V. We just write it as V(x) because we're usually going to use it that way later. Any unitless predictions you get to any actual question are necessarily unchanged by those choices of coordinate systems (whereas if they have units then they are measured in one of the coordinate systems).
So I would say that (a,(b,c)) and ((a,b), c) are just two different coordinate systems for R^3. But necessarily any math you do with R^3 can't depend on the choice of coordinates. There is probably a way to write that by putting something like "units" on every term and then expecting the results of any calculation to be unitless.