It's not necessary, you can drop it. If there's a possessive pronoun (in English those are possessive adjectives I believe) before the noun, the article is not necessary. If they come after the noun though (like the pronouns "dele", "dela", "deles", "delas"), the article is necessary.
You can say "Meu aniversário é em julho", but you have to say "O aniversário dela é em julho".
Aniversário serves for both. I suppose Duo is being picky because when we say "Meu aniversário é em julho" the most common thing to think is that it's the birthday... If it wasn't, the person would have to clarify saying "Meu aniversário de casamento é em julho". But there are a lot of people who just says "aniversário" without specifying what it is, then the other person has to ask "Aniversário de nascimento ou de casamento?". Saying "Nosso aniversário é em julho" would help giving the clue that it's a marriage anniversary, for example (not always though). But with context, you should know what type of "aniversário" one is talking about.