Well, it's also "in" in English, so it maps pretty cleanly here; I think the question may have been more about how being "in" a month sounds weird when you consider "in" in a kind of literal, "inside something" sense. In other words, more about how to use prepositions literally in Esperanto, rather than translating literally from some other language. Perhaps in that sense one might expect "dum" here instead.
However, I don't think it's about being as literal as possible in this sense. It's more about picking the one that is the most logical—not based on any other language, but just based on the concept you're trying to express. In that sense I think both "dum" and "en" would work here. And indeed, "en" ends up being the one that is used in this context.
When your talk about something that spans several days, like "weekend", "summer" or "vacation", "during" is more like "spanning the entire" or at least "spanning several days of"
"I was drinking in the weekend" -< "i was drinking at some point in the weekend". "I was drinking during the weekend" -> "i was drinking the whole weekend"
"my birthday is during march" would mean that your birthday is starting in march, and ends some number of days later in march.
"my birthday is in march" would mean that your birthday is on a single day of march.
"Is"?? What do you mean? Possibly you meant that your birthday is in March. "Estas" can mean "it is" where there is no "actual it", as Salivanto calls it, so "It is cold outside" is "Estas malvarme ekstere." But if you want to say, "It is", meaning "My birthday is", then you need a word for "it", "ĝi" in Esperanto.