I think those two can be the same in English, "a" and "per." Stergi3 explains that some prepositions act as adverbs: https://forum.duolingo.com/comment/17589977 I'm just a learner of modern Gk, so we'll need confirmation from a moderator or Gk speaker, but my understanding is that accusative is used in expressions of time measurement, as in τρις φορές την ημέρα, "three times a day" // "three times per day." Here in this instance above we have adverbial ανά. Here's a link to some examples of idiomatic use of preposition and adverb ανά: http://www.asteri.ws/en/gramm2/prep-en.html One example is ανά τον κόσμο, which is preposition with accusative = σε όλο τον κόσμο. Prep example: μία ώρα ή δύο την ημέρα, "one or two hours per day." https://el.wiktionary.org/wiki/ανά
If we're talking about pills, for example, you might be taking two of them at the same time, so that'd be "once a day". "Twice a day" means that you have to repeat the process of taking at least one pill one more time. Your translation's about frequency, whereas the translation given, which is the correct one, focuses on quantity.