It needs to be understood that "canonical" was really introduced in the context of search engine optimization and is supported by web search engines. In essence, if a landing page would have a "canonical" link to the DOI RL, the content of the landing page would be indexed under the DOI URL.
For browsers, the desired behavior would be to bookmark a page that has a "canonical" link under the canonical URL instead of the page's URL. At this time, to the best of my knowledge, this is not supported by browsers. However, it seems to me that it should not be impossible to achieve this, if, for example, the Persistent Identifier community (DOI, handle, PURL, ark, ...) would lobby for it. Alternatively, one could write an RFC and define another link relation type specifically for this relation type between landing page and persistent identifier. And lobby browser manufacturers to implement the "right" bookmarking behavior for it. But, the difference between "canonical" and this special-purpose relation type might be too subtle to make it through IANA registration scrutiny.
In the context of scholarly articles, if you're reading an article, the references will be linked via DOI, e.g. here: http://journals.plos.org/plosone/article?id=10.1371/journal....
Where this all gets interesting is non-traditional scholarship, e.g. sharing on social media where there are lots of 'publishers' (twitter, reddit) and they don't particularly care about best practice from the scholarly publishing industry. How do you get someone to share the persistent link? You can't force them. It's not a solved problem, and probably never will be.
Herbert Van de Sompel is working in this area http://public.lanl.gov/herbertv/home/ with things like canonical link headers that indicate the canonical URL of a given page. Here's a presentation he did on the subject, which he calls 'signposting the scholarly web': https://www.force11.org/video/signposting-scholarly-web