What is it called when someone presents two choices which are the same to emphasize the importance of the option?
The form “x or y” (where x and y are possible statements or actions) is called a dilemma.
When other options are (deliberately) omitted (implying that the mentioned two are the only realistic options), it is called a false dilemma, which is a type of informal fallacy.
When x and y are the same (implying that only twice-mentioned option is the only realistic option), the form becomes “x or x”. I suppose that one could call this an isodilemma.
I admit to making this term up, but I believe it fits.
And I agree that the literary device is common enough that it deserves a fancy name.
(I also considered, but dismissed, *a monolemma.)