This question popped in our exam in which you had to choose which particle was right. The question was:
with に or まで as the options. まで didn't make sense for me and I don't know if going with に is possible.
Yes it's perfectly possible to use the same particle twice or more in a sentence and even in the same clause, and it is very common.
There are some restrictions e.g., I don't think you can have the object marker を more than once within a single clause, but then を only has one function (almost). に is a particle with many different functions.
You are correct that the missing particle is に and this does not cause any ambiguity or difficulty in understanding.
(I) gave flowers to mum on (her) birthday.
Yes it is possible and repeated use of に in your sentence would be the most correct and natural way.
誕生日に on my Mom's birthday (implied to be Mom's birthday based on context)
母に to my Mom
花をあげました I gave flowers
There is no rule that a particle cannot be repeated in a sentence and your sentence can be further extended.
On my birthday, I gave Mom flowers and I gave Dad flowers and I gave my little brother flowers and without giving Grandma flowers I ran away.
It is possible to have. に has different meanings after 誕生日 and 母.