Sonnet, Dramatic monologue
Duffy's "Anne Hathaway" is a sonnet spoken in the voice of Anne Hathaway, Shakespeare's wife. Because Duffy imagines the speaker as one distinct character, we can call this poem a dramatic monologue. A dramatic monologue doesn't have to follow any set form, it just has to be spoken in the voice of a specific persona, real or imagined.
"Anne Hathaway" does have a form, though. It's a sonnet! It follows the most basic sonnet rule: it has fourteen lines. But that's pretty much it – otherwise, it breaks a lot of rules. For example, it has no formal rhyme scheme and its meter is only roughly iambic pentameter.
Side note: iambic pentameter is a fancy way of explaining the consistent da-dum, da-dum, da-dum, da-dum, da-dum (yep, five times!) rhythm of the lines. A perfect example is line 5 of Shakespeare's Sonnet 116.
O no! It is an ev-er fix-ed mark (sonnet 116, line 5, where the italics show the stress)
Compare this to a line from Duffy's poem:
drib-bling their prose. My liv-ing laugh-ing love – (12)
Close, but no cigar!
Duffy definitely takes a lot more liberty with the form than Shakespeare does. But her poem does include what is perhaps the signature Shakespearean sonnet feature: the couplet. Duffy's sonnet, like all of Shakespeare's, ends with a strong couplet (a pair of rhyming lines). Adds a nice punch, we think.
If you're interested in learning more about sonnets, check out what we have to say about them in our poetry glossary. If you want to know more about Shakespeare's sonnets in particular, we've got some handy study guides just waiting for you.