Does this translate into English as 'day AFTER day' or as 'day BY day'? They have subtly different meanings in English. The first tends to have negative connotations, whereas the second suggests something positive. Thus we might say: I slog away on Duo day after day, but I seem to learn nothing. Alternatively: Day by day I'm getting that little bit better...
"I clean up after my roommate day after day" is frustration about how often you do a thing. You may also hear "I do this day after day, after day, after day!" from very frustrated people.
There was a TV show in the 90s with a theme song, "Step by step, day by day, a fresh start over, a different hand to play..." This is something you're improving upon daily, or often.
"Duolingo has become part of my day to day life," means it's something uninteresting you do daily.
I would not use "day to day" as WisalCh did, because "day to day" doesn't imply gradual improvement, just it's something you do every day. Maybe it's a regional difference? Now that I've rambled my English usage, which version works for this translation?
There are two or at least one and a half reasons why Duo may be rejecting "Day to day" as a translation for "Tag um Tag."
"Tag um Tag" means "day after day."
(or just 1.5) In Standard English "Day to day" is written with hyphens: "day-to-day"—which corresponds nicely with the fact that the expression corresponds to the single German word "täglich."