Well, if you look at Déardaoin", it actually contains the Dé, it's been written into the word; you'll never see Dé Déardaoin. It's necessary when saying things like "This Tuesday," or "On Tuesday."
Mar shampla Dé Luain means "On Monday (i.e., this coming Monday) but Ar an Luan means "On Monday" (i.e. On Mondays)
Well, most Western European languages get their names for the days of the week either directly or indirectly from Latin. In the Germanic languages, the planets/gods were replaced with Germanic equivalents, dies Martis -> Tiw's Day (Tuesday), for example. Irish directly borrowed several of the weekdays from Latin. Spanish, of course, inherited the names from Latin
I know this is not directly linked to the question, but I wish Duo signaled spelling errors in exercises where we must write what we hear... I wrote "Dé Márt", wondering if it wasn't "Dé Máirt", and it said it was correct and it meant "Tuesday". Yeah, but what about my spelling error :-( For me, broadly remembering the names of the days is not the problem, precisely remembering their different forms and when to use them is...